I use keepass too. I keep my database in dropbox and use the android dropbox and keepass clients on my android. Logging into an app or website involves opening dropbox, clicking on the database[1], entering my password, choosing the site, and clicking on "copy password to clipboard." It's a few extra steps, but it's not that much of a hassle.
[1] I find this easier than opening keepass and selecting the database from dropbox for some reason that might be as simple as dropbox having an easier to spot icon.
You can also use the favorite feature on Dropbox to keep a fresh copy of the database on your phone and have KeePassDroid remember that location. Then your flow is 1) open KeePassDroid 2) enter password 3) select site 4) copy/paste
[1] I find this easier than opening keepass and selecting the database from dropbox for some reason that might be as simple as dropbox having an easier to spot icon.