is there any lake located in the former island?
Yes. Lake Tegano
Rennell Island
Official Solomon Island maps name this lake Te Nggano, while locals refer to the lake as Tegano or Tungano or Big Water. The lake is in length and wide, with an area of , which occupies 17.6% of the total area of Rennell Island. 
when was battle of rennell island fought? 29–30 January 1943 between whom was it fought? United States Navy and the Imperial Japanese Navy where did the battle take place? In the South Pacific between Rennell Island and Guadalcanal in the southern Solomon Islands where is the former island located? In the Solomon Islands did this place experience any earthquake? Yes. On 2 April 2007 and on 6 February 2013 where did it take place? First earthquake northwest of the island's capital, Honiara and south-east of the capital of Western Province, Gizo and second in the Santa Cruz Islands was this place ever colonized? Yes. By Germany and Britain did the former experience a republican era anytime before? UNANSWERABLE
wiki:13795269
Rennell Island [SEP] Lake Tegano
Official Solomon Island maps name this lake Te Nggano, while locals refer to the lake as Tegano or Tungano or Big Water. The lake is in length and wide, with an area of , which occupies 17.6% of the total area of Rennell Island. The lake is located at East Rennell, the southern portion of Rennell Island, in the central basin which was the old lagoon. The hard bottom is covered by several meters of suspended mud. The lake has an elevated salt concentration being kept by a subterranean duct system which connects it with the sea. The water depth is and consists of a mixture of brackish fresh and salt water.