If cleaning doesn’t solve the issue (I would try again with alcohol), then you will probably have to [guide|43769|open it up] and take a direct look at the Home Button to see what is causing the issue. These devices, while certainly DIY repairable, are not fun to play with so check out the guide first to see if you want to do this. Open up the device first, inspect to understand what is causing the damage and then order the parts you need. That way you can always decide to pursue the repair or not, depending on the cost of parts (this is an old device after all).