How can I work around a broken faucet handle?
That is the stem, but looking at your pictures it appears the faucets are 1/4 turns. So if you can turn it more that 1/4 of a turn and not get water, you need to turn it the other way. If the other way doesn't work you'll need to replace it, because that's all the knob/handle would do. You may have loosened it and need to turn it back to where it was to start. Guessing from photo on the 1/4 turn, if they aren't you may need to just turn more or try the other way.

You can also remove the screw holding down the other handle and use the same handle on both.