I think "the white slot thing" is the bridge saddle. The part that is still in the slot is a shim that was installed to adjust the bridge height.

There should be some small notches cut into one of the edges off the piece that fell out. Even if notches were not actually cut, the strings will have left marks on the material. These will match the size of the strings. The largest string was the bass E, and the notch for it is the widest one. Put the saddle into the slot so that the notch for the bass E is where the bass E string will be installed. The treble E string is much smaller in diameter, so the mark left by it will be much narrower than the one for the bass E.

In future, get your new strings [u]before[/u] you take off the old ones, and swap them out one at a time. Our own Kevin Taylor has an excellent tutorial >here<.