Apple uses a proprietary encryption algorithm to generate a unique key for each Touch ID fingerprint sensor that guarantees it's paired to a specific logic board. When you lose the original home button you've lost that key; it can't be read or duplicated.
As @flannelist mentioned, the only way to have a working Touch ID is to buy a replacement motherboard ***and*** its paired home button. Apple alone can replace a home button; they keep a very tight grip on their encryption algorithm and don't let anyone but authorized service centers even generate the key and program it into a button.
So just to reiterate; if you didn't buy the home button along with the motherboard, then you won't get Touch ID working again, period. There are no aftermarket buttons, parts or programmers that will make it work.
A trip to the Apple store is the only way to get a Touch ID repaired. I seem to recall you can't just ask them to replace the home button; that doesn't seem to be one of their services. If, however, you have the screen replaced, then they'll also replace the home button and you'd get your Touch ID back. That assumes they'll even work on it since the logic board has been replaced; if I'm not mistaken they tend to refuse to work on phones that have been repaired by non-Apple persons.
There's also the self-repair program, assuming it supports your phone. Of course, you have to buy the parts from Apple and rent their repair equipment, but it might be possible under those circumstances.
Alisha, have I stated these last two paragraphs correctly or am I just blowing smoke out my a$$?