TouchID is code locked to the CPU, NAND and Baseband (it varies depending on the model). that means if you damage a Home button, the only way to make it work is to have the screen replaced at Apple so that they can re-program a new button. To be honest, I don't know if Apple is still replacing screens on the iPhone 5S as of today.
Some micro-soldering repair shops may be able to repair the Home button flex cable depending on where and how much it is damaged however there is no guarantee of success for this option.
From a DIY repair standpoint, you either live without Touch ID or get a new logic board/Touch ID combination. However, the cost of that is the same or higher than a replacement iPhone 5S that you could get on secondary markets (Craigslist, Kijiji etc).