Need some help with exposure

Firstly, I'd love it if you'd hop over to Sarah's and vote for my photo (second one down).  It was picked in the top 4 for June.  Thank you!  But make sure to come back!  :)
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Around 4:30 this morning we were awakened with IV machines beeping ... and beeping ... and beeping.  Once Alex's nurse got the machines quieted, it was a little hard for me to get back to sleep.  Especially once I noticed how cool the moon looked in all its fullness.  So I broke out my camera.  Only problem was I had no tripod.  But I did have a little red cooler with a flat top, so I balanced it on the window sill and starting playing.  Alex's nurse was in and out a couple times while I was playing and probably thought I was nuts.

I was trying to get a shot of the full moon over the city of Pittsburgh.  But I was having trouble with exposure.  If I kept the shutter open the correct amount of time to have a well-exposed moon with all the details, you could barely see the city lights.  And if I kept the shutter open long enough to see the city lights, I had an over-exposed moon with no detail.

ISO 100, f/11, 5 second shutter speed

I love the city lights in this photo but the moon is just a white glowing blob.  Not good.

So I took a few photos of just the moon ...

ISO 100, f/11, 1/20s

This turned out pretty well considering I was balancing the cooler on the window sill and the camera lens on the lens cap to raise it up a bit.  

Since I didn't end up with the shot I was after, I made a composite ...

What do you think?

What I need are tips on how to expose something like this so I can get the bright city lights and a well-exposed moon.  Or isn't that possible?  I know people use black foam board and stuff to partially expose parts of their frame, but I don't have anything like that here in our little hospital room.  And I'm not sure my brain would have worked that well at 5 a.m. anyway.  I have my tripod now so I might be able to play more.  Except that I am completely exhausted and would love about 18 hours of uninterrupted sleep.   I know that's not possible in a hospital though.

So I doubt I'll play again while we are here.  Except to photograph the fireworks tomorrow night.  I'm told we will have a pretty good view of the fireworks display.  One of the doctors said we should sell tickets to our room.  :)  I hope Alex feels well enough to enjoy them with me!

                

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