Pull up your screen. iFixit has a guide that shows how to pry it up. Once your screen is up, you will see that the home button is secured on in two tiny spots with basically double sided tape. Remove the old adhesive, make sure you get all of it. Clean up the bracket and screen so there is no dirt or residue. Then apply some red tape or your favorite adhesive. Make sure you set it firmly in place. Then reseal your screen and you are done.
If it were me, I'd buy a new digitizer that includes a home button assembly. Chances are that when you pry up your screen, you will scratch it and maybe crack it. An iPad 2 digi with small pieces isn't very expensive and it will come with the home button, bracket and tape already in place. It would make very quick work of this job.