This question on Baali mentions that the arrows used by Shri Rama used to kill Baali were also used to kill Kumbhakarna. Shri Rama used a 100 arrows and even those could not kill KumbhaKarna.
What is the story KumbhaKarna's death? How did he die?
Yes, It is true that those arrows which killed Vali also couldn't do any harm to Kumbhakarna. Here is how Valmiki Ramayana describes Kumbhakarnas Death::
दर्शयेक्ष्वाकुशार्दूल वीर्यम् गात्रेषु मे लघु | ततस्त्वां भक्षयिष्यामि दृष्टपौरुषविक्रमम् || ६-६७-१५२
"O faultless excellent Rama born in Ikshavaku dynasty! Show your prowess on my limbs. After seeing your strength and prowess, I will be devouring you."
स कुम्भकर्णस्य वचो निशम्य | रामः सुपुङ्खान्विससर्ज बाणान् | तैराहतो वज्रसमप्रवेगैर् | न चुक्षुभे न व्यथते सुराइः || ६-६७-१५३
Hearing the words of Kumbhakarna, that Rama released plumed arrows. Even after struck by them, whose speed was equal to a thunderbolt, that demon was neither shaken nor afflicted.
यैः सायकैः सालवरा निकृत्ता | वाली हतो वानरपुङ्गवश् च | ते कुम्भकर्णस्य तदा शरीरं | वज्रोपमा न व्यथयां प्रचक्रुः || ६-६७-१५४
Those arrows, which chopped off the Sala trees and killed Vali the foremost of monkeys, could not torment Kumbhakarna's body which was like a thunderbolt.
स वारिधारा इव सायकांस्तान् | पिबन् शरीरेण महेन्द्रशत्रुः | जघान रामस्य शरप्रवेगं | व्याविध्य तं मुद्गरमुग्रवेगम् || ६-६७-१५५
Sucking those arrows with his body, as mountains suck up torments of water, that Kumbhakarna, flourishing his hammer with terrible speed, hindered the tremendous speed of Rama's arrows.
ततस्तु रक्षः क्षतजानुलिप्तं | वित्रासनं देवमहाचमूनाम् | व्याविध्य तं मुद्गरमुग्रवेगं | विद्रावयामास चमूं हरीणाम् || ६-६७-१५६
Then, flourishing that hammer which was smeared with blood and which can frighten the great army of celestials, in terrific speed, that demon scared away the army of monkeys.
वायव्यमादाय ततो वरास्त्रं | रामः प्रचिक्षेप निशाचराय | समुद्गरम् तेन जहार बाहुं | स कृत्तबाहुस्तुमुलं ननाद || ६-६७-१५७
Thereupon, taking a great missile called Vayavya, Rama hurled it on the demon. By that weapon, he chopped off Kumbhakarna's arm along with the hammer. With his arm chopped off, Kumbhakarna roared tumultuously.
स तस्य बाहुर्गिरिशृङ्गकल्पः | समुद्गरो राघवबाणकृत्तः | पपात तस्मिन् हरिराजसैन्ये | जघान ताम् वानरवाहिनीं च || ६-६७-१५८
That Kumbhakarna's arm, identical to a mountain-peak, which was chopped off by Rama's arrow, fell along with the hammer on that army of Sugreeva and killed that regiment of monkeys.
ते वानरा भग्नहतावशेषाः | पर्यन्तमाश्रित्य तदा विषण्णाः | प्रपिडिताङ्गा ददृशुः सुघोरं | नरेन्द्ररक्षोऽधिपसंनिपातम् || ६-६७-१५९
Those monkeys who had escaped being broken and slain by that arm, though dejected with their tormented limbs and taking recourse to the sides, witnessed a highly terrific encounter between Rama and Kumbhakarna.
स कुम्भकर्णोऽस्त्रनिकृत्तबाहुर् | र्महासिकृत्ताग्र इवाचलेन्द्रः | उत्पाटयामास करेण वृक्षं | ततोऽभिदुद्राव रणे नरेन्द्रम् || ६-६७-१६०
Having an arm chopped off by the arrow like a mountain-peak chopped off by a gigantic sword, that Kumbhakarna with his another arm, pulled up a tree by its roots and then ran towards Rama the lord of men in that battle-front.
स तस्य बाहुम् सह सालवृक्षं | समुद्यतं पन्नगभोगकल्पम् | ऐन्द्रास्त्रयुक्तेन जहार रामो | बाणेन जाम्बूनदचित्रितेन || ६-६७-१६१
By his arrow, which was made variegated by gold and furnished with a mystic spell of Indra used for charming it, Rama chopped off Kumbhakarna's remaining arm, appearing like the coil of a serpent along with his uprooted palm-tree.
स कुम्भकर्णस्य भुजो निकृत्तः | पपात भूमौ गिरिसंनिकाशः | विवेष्टमानो निजघान वृक्षान् | शैलाञ्शिलावानरराक्षसांश् च || ६-६७-१६२
That Kumbhakarna's arm, which appeared like a hill, was chopped off and fell down on the ground. Wallowing hither and thither, it dashed with trees, rocks, monkeys and demons.
तम् छिन्नबाहुम् समवेक्ष्य रामः | समापतन्तम् सहसा नदन्तम् | द्वावर्धचन्द्रौ निशितौ प्रगृह्य | चिच्छेद पादौ युधि राक्षसस्य || ६-६७-१६३
Seeing Kumbhakarna with his arms chopped off, abruptly with a roar, coming upon him and taking two sharp arrows with a shape of a half-moon each, Rama chopped off the feet of the demon in that battle.
तौ तस्य पादौ प्रदिशो दिशश्च | गिरेर्गुहाश्चैव महार्णवम् च | लङ्काम् च सेनाम् कपिराक्षसानां | विनादयन्तौ विनिपेततुश्च || ६-६७-१६४
Creating a resound everywhere in all directions, even in hill-caves, in the great ocean, in Lanka as also in the armies of monkeys and demons, Kumbhakarna's feet fell down.
निकृत्तबाहुर्विनिकृत्तपादो | विदार्य वक्त्रं वडवामुखाभम् | दुद्राव रामम् सहसाभिगर्जन् | राहुर्यथा चन्द्रमिवान्तरिक्षे || ६-६७-१६५
Widely opening his mouth like the mouth of a submarine fire and roaring, Kumbhakarna whose arms and feet were cut off, ran (with thighs) quickly towards Rama, like Rahu the seizer-demon going to seize the moon in the sky.
अपूरयत्तस्य मुखं शिताग्रै | रामः शरैर्हेमपिनद्धपुङ्खैः | स पूर्णवक्त्रो न शशाक वक्तुं | चुकूज कृच्छ्रेण मुमोह चापि || ६-६७-१६६
Rama filled up Kumbhakarna's mouth with sharply pointed arrows, having shafts covered with gold. With his mouth full of arrows, Kumbhakarna was unable to speak. He moaned with difficulty and even became unconscious.
अथाददे सूर्यमरीचिकल्पं | स ब्रह्मदण्डान्तककालकल्पम् | अरिष्टमैन्द्रं निशितम् सुपुङ्खं | रामः शरं मारुततुल्यवेगम् || ६-६७-१६७
Thereupon, that Rama got hold of an arrow spelled with a missile presided over by Indra the lord of celestials, effulgent as sun's rays, resembling the rod of Brahma the lord of creation as also the destructive Kala, the Time-Spirit, and having its speed equal to the wind.
तम् वज्रजाम्बूनदचारुपुङ्खं | प्रदीप्तसूर्यज्वलनप्रकाशम् | महेन्द्रवज्राशनितुल्यवेगं | रामः प्रचिक्षेप निशाचराय || ६-६७-१६८
Rama discharged against the demon, that arrow, whose shaft was inlaid with diamonds and gold, which was shining as the dazzling sun and fire set ablaze, and which vied with the speed of Indra's thunderbolt.
स सायको राघवबाहुचोदितो | दिशः स्वभासा दश सम्प्रकाशयन् | विधूमवैश्वानरदीप्तदर्शनो | जगाम शक्राशनितुल्यविक्रमः || ६-६७-१६९
That arrow, propelled by Rama's arm, with a terrific aspect like the smokeless fire, having a formidable energy of Indra's thunderbolt and illuminating the ten quarters with its own splendour, proceeded forward.
स तन्महापर्वतकूटसंनिभं | सुवृत्तदंष्ट्रं चलचारुकुण्डलम् | चकर्त रक्षोऽधिपतेः शिरस्तदा | यथैव वृत्रस्य पुरा पुरन्दरः || ६-६७-१७०
That Rama slashed Kumbhakarna's head, which was looking like a huge mountain-peak, having well-rounded tusks and with charming and quivering ear-rings, as like Indra the destroyer of strong-holds, in the past, chopped off the head of Vritra, the demon of darkness and drought.
कुम्भकर्णशिरो भाति कुण्डलालम्कृतं महत् | आदित्येऽभ्युदिते रात्रौ मध्यस्थ इव चन्द्रमाः || ६-६७-१७१
The large Kumbhakarna's head, adorned with ear-rings, shone like the moon being in the middle, when the constellation, Punarvasu (presided over by Aditi the mother of gods and consisting of twin-stars) has risen at the night.
Kumbhakarna, the Rakshasa had never been defeated in battle before. On the battlefield, Kumbhakarna kills innumerable vanara and eats many of them. He then restricts God Hanuman and Angada. Sugreeva barely escapes death. God Brahma appears before God Rama and advises worship of Goddess Durga, though it is an odd time (shraadha) and a dark fortnight [Ref: Mahabhagavata Upapurana 41.13-14, 42.19-23].
God Rama does as told and obtains Her blessings. He goes on to kill the mighty Kumbhakarna after a fierce battle, thanks solely to the blessings of Goddess Durga [Ref: Valmiki Ramayana 6.60-68].
Source: 'Essence of the Fifth Veda' by Gaurang Damani page 45