No English translation is available. The complete book is available in sringeri math in sanskrit in two volumes. Though it is not related to your question, some background is necessary for the book, as your statement that Laghu Chandrika is a refutation to tarangani is not strictly correct. It not only includes refutation of the Tarangini of Ramatirtha and also Nyayamrtakantakoddhara, which was the latest text at that time. The following history may be useful to you.
Please note that the advaita dvaita controversy has been going on for more than seven centuries. General refutations of each text has been going on. However, Madhusudhana Saraswati was the first to start what is known as text-by-text refutation.
The masterpiece of Vyasatirtha (15th century) is Nyayaamrta. This is a refutation of advaita based on the methodology of Navya-Nyaya. This is the still considered the gold standard for Dvaita philosophy of Madhava Acharya.
Anandagiri Tattvaloka (fourteenth century A.D.), Vidyaranya (fifteenth century A.D.) Pancadasi, Nrsimhasrama (sixteenth century A.D.) Advaitadipika, Appayyadikshita’s (sixteenth century) Madhvatantramukhamardana all wrote against dvaita principles.
But the key work that was a text-by-text refutation of Nyayaamrta was Madhusudhana Saraswati's Advaita Siddhi.
The origin of Advaita Siddhi as a retort to Nyayamrta started a rally.
Ramatirtha (circa 1550-1620 A.D.), a disciple of Vyasatirtha, wrote Tarangini to refute Advaita Siddhi.
This Tarangini of Ramatirtha was refuted by Balabhadra (c. 1600-1700 A.D.), a disciple of Madhusudana. He wrote Sidhhivyakhya on Advaitasiddhi, which is a verbatim reply to Tarangini.
Refuting Siddhivyakhya there originated another commentary called Nyayamrtakantakoddhara attributed to Anandabhattaraka.
Brahmanandasarasvati (c. 1600-1700 A.D.) wrote the commentary Gurucandrika and its condensed form Laghucandrika on Advaitasiddhi, in refutation, mainly, of the Tarangini of Ramatirtha and also Nyayamrtakantakoddhara.
Vanamalimisra (c. 1650-1720 A.D.) wrote two commentaries namely Taranginisaurabha and Nyayamrtasaugandhya. These are in refutation of Gurucandrika and Laghucandrika.
Vitthalesopadhyaya (c. 1700-1800 A.D.) wrote a commentary on Laghucandrika known as Vitthalesopadhyayi. This refuted Vanamalimisra commentaries.
The controversies still continue. There are also refutations and counter refutations of other texts. For example,
Sankarapadabhusanam of Raghunathasuri (1821-1859 A.D.) refuted the arguments of the
Tatparyacandrika of Vyasatirtha. Rangappacarya (1820-1891 A.D.) wrote Candrikabhusanam in refutation of Sankarapadabhusanam. Ramasubrahmanyasastri (1850-1920 A.D.) wrote Candrikakhandanam. Satyadhyanatirtha (1913-1942 A.D.) wrote Candrikamandanam in refutation of Candrikakhandanam. Thus, this goes on.