At Srirangapatna, you have couple of good options for staying overnight. The KSTDC Mayura River View, Cute Resorts, Young Island Resorts etc are all good and quite well maintained.
Srirangapatna is an island town encircled by river Cauvery. The temple of Lord Sriranganatha is said to have been built by the Chieftain Thirumalaiah in 894 AD. This was once the capital of Hyderali & his son Tippu Sultan, "The Tiger of Mysore". In the 18th century, Marathas attacked Srirangapatna & forced heavy payments. Hyder drew back the Marathas & saved the Kingdom. In 1799 Tippu fought a fierce Battle (Battle of Mysore) against the British & fell bravely defending his capital. Srirangapatna remains in a ruined state recalling the Battles fought & the past glory.
If you wish to visit Srirangapatna in a conducted tour, there is a daily trip covering Srirangapatna and Mysore by KSTDC.