Buzzing/whirring noise in engine kills battery while car is off

Noticed yesterday car was having trouble starting. Almost like it wouldn't even recognize that the key was being turned. After some fiddling with the key and brake it would finally start like normal (no prolonged cranking or anything). But there was a new noise coming from the engine that we haven't heard before. Like some kind of whirring sound that turned off after a minute or two.

Now today I went to start it to see if I could reproduce the noise. At first it did the same thing by not starting until I had fiddled with it a bit. However, now, when I turned the engine off and removed the key, there was still some noise on the right, lower, side of the engine (facing the engine) that kept going. Almost like some motor going on down there. After about 5 minutes of that it finally killed the battery completely. So I went to jump it and as soon as I connected the cables that motor noise kicked back on.

Any ideas what that sound is? The attached image shows about where I hear the noise coming from.