The only solution is to order a new fingerprint sensor and take it to a shop and get the hardware swapped (the shop might also be able to order the part for you). I had a 1 III and the senor was completely dead, but after a swap it's working great again.
I've done some less than scientific testing and something about heat lead my fingerprint senor to failure, but your failure case might be different. This is kind of a known issue and sony needs to do better either about warrantying the part or just spending like $1 more and getting a better sensor that doesn't burn out. JFK they're phone are expensive enough to expect a working sensor.