Jen, you might want to smother that corner of the bin with a wet cotton cloth (ripped up t-shirt). Just leave that area alone for a couple of weeks and see if the buggers die off.

Andrew

Make a trap.

Use a narrow mouthed plastic bottle, a water or pop bottle will work. Drill three little holes in the lid. Cut a third off the top of the bottle. Invert the top into the bottom of the bottle. Use the food they are attracted to as bate. Place the bottle on top of the bin. Or if possible inside the bin.

When you have a lot of flies in the trap take the bottle outside and dump it. Then rebate the trap.

Keep a vacuum handy when you open the bin and sweep them up as they fly out. Just crack the bin enough to make them come through where it will be easy to catch them without chasing them. Chasing them in the air will just tick you off more. You definitely don't want to put the vacuum too close to the bedding! Do this in addition to what Andrew mentioned or use cardboard. If I get flies (nearly always seem to come in on bananas), I use cereal box cardboard to cover the bedding. If the flies come out, they will come out around the edges where it is easy to catch them with vacuum hose. Tapping the cardboard will make several come out at once.