The shloka mentioned in the question is as follows.
प्रस्थे हिमवतो रम्ये मालिनीमभितोनदीम् |
जातमुत्युज्य तं गर्भ मेनका मालिनीमनु'॥
Malini is the river originating in Shivalika mountain range in the state of Uttarakhand (India) and is flowing through Uttar Pradesh merge with holy river Ganga flowing through cities Bijnor and Najibabad . Actually there are three rivers of similar names flowing side by side called as Milini , Malin and Malini. River Malini branches out from Malin river and merge in Milini river , the in between flow of the water is known as Malini river.
Here is the Hindi Information about Malini River
In Hinduism scriptures besides Mahabharata Malini river is also mentioned in Valimiki Ramayana -Book II : Ayodhya Kanda -Sarga 68- Shloka 12. Where its location is also mentioned .
न्यन्तेनापरतालस्य प्रलम्बस्योत्तरम् प्रति |
निषेवमाणास्ते जग्मुर्नदीम् मध्येन मालिनीम् || 2.68.12 ||
Those messengers went on touching Malini River, flowing between the passing Aparatala Mountain and the northern end of Pralamba Mountain.
In his book History Of Ancient India (a Neriver Version) : From 7300 Bb To 4250 Bc , author J.P. Mitta mentions that the Malini is known now as Ramganga.
The river malini after rising from Gharwal hills flowed towards the
west of Najibabad in district Bijnor of U.P. and then met the river
Ganga . The river Malini is known as Ramganga.
As reported in this News 18 article - April 2017 . River Malini is getting extinct now a days , most probably because of climate change , drying due to reduced base flows and heavy extraction.
And finally as your inquiry about its current location , here is Google Map of current Malini river
, where we can trace the precise location of river and the kanva Muni Ashram by navigating on map. At Kanva Ashrama the river is known as Milini and the Malin river then one branch of Malin river is called as Malini river . The river seems to be getting smaller as it flows and merges with some other river named Milini.