You have the correct Lenovo software installed, I'll assume? Go into the Tutorial in the FPR app and you can actually test the scanner at will to see if it's actually working.
Go into the machine BIOS and clear the security chip and then attempt to reenroll fingerprints again; it's possible that the security hash table is corrupted or something of the sort.
It could potentially be that the internal connector is coming unseated, but an X41 Tablet is probably more complicated to disassemble than most people are willing to try (unlike most ThinkPads where most components are easily user-serviceable). If you're technical, I'd recommend going to the Lenovo site and looking for the Hardware Maintenance Manual for that machine model type number because it's worth a try.
If none of that works, it's probably just a hardware failure. This is a pretty old machine, and the build quality is obvious if it's lasted you this long with no problems! Good excuse to upgrade to an X201 Tablet. :)