Hi Oiid,
If we compare the picture of your home button with one that's available for sale on the internet, it looks to me like the metal dome that gives the button its "click" is what you're missing. You're not going to be able to buy that piece by itself, but if you go online and buy a replacement button, you should be able to remove that dome from the new one and glue it onto your existing button. Here's the piece you need.
[image|2950901]
You can buy the button right here on iFixit if you'd like.
[product|IF124-013|iPhone 5s Home Button Assembly|new_window=true]
Offhand I don't see anything else wrong with your button, so hopefully that will fix it and let you keep your Touch ID.