From the Bitcoin Android wallet app?  or from Blockchain.info for Android?   or something else?

That's a long time for a transaction to hang out there without getting a confirmation.  The original sender must be re-broadcasting it periodically (which is what the client is supposed to do).  

But because it has an output below 0.01 BTC and it has no fee, not all nodes are relaying it.  So you still might be able to spew out a double spend that would get relayed.

So if you have the private key for the address from one of the inputs for that send transaction you can import that address into the Bitcoin-Qt client, or Multibit, or Electrum, etc. and try to spend from one of those.  You'll have the same problem if the amount is below 0.01 BTC  and you don't pay a fee though so make sure to pay the freight.

[Update: Just remember though, with Bitcoin-Qt if that is a failed double spend attempt there's no way to delete the offending transaction from the wallet (without using the third party tool, pywallet, to perform wallet surgery).  So don't do this with your normal wallet.dat for Bitcoin-Qt, or at least make a backup right before so you can restore the old one if the double spend attempt never confirms.]

That may not even be necessary.

If the original transaction didn't get relayed very well because it was below 0.01 BTC and no fee then a node running Bitcoin-Qt wouldn't know about the original transaction and thus wouldn't have a problem creating a spend transaction.  So simply importing one of the private keys used in that transaction to Bitcoin-Qt and spending it (with a fee) might be all that is needed.

But seeing your suggestion to use Brainwallet.org ... that is probably an even better idea than trying to get a Bitcoin-Qt set up just for this one transaction.

Success! Thank you guys for the help. It took a little digging, but in hindsight, the solution is relatively straightforward.

I didn't start with any backups, but I did spend everything I could to another wallet. I wouldn't do it with any amount I'd be afraid to lose.

Exporting the private keys from the Android wallet app was simple enough. The only issue here is that you can't get them in plaintext. (n.b.: This is a security feature as basically any other app would have access to them--if you have root on your device, I believe you can get the wallet.dat file, but that wasn't an option here.) Give it a password so that you can export it. It'll create a file called "bitcoin-wallet-keys-YYYY-MM-DD". Move that file onto a computer via SD card or your file transfer mechanism of choice.

Unfortunately, most of the suggestions below couldn't obviously handle an encrypted key file. Brainwallet, pywallet, electrum, Bitcoin-qt? No luck. MultiBit? Jackpot.

Give your "bitcoin-wallet-keys" file a ".key" extension and import it with MultiBit. There's a field for the password. I've got no idea how it's encrypted, but obviously it's the same as the wallet app. From here, you're free to spend the coins as MultiBit is ignorant of the transaction that's supposed to be propagating through the network (for the past #@%$#@ing week!)

All that said, I'm not sure that I actually had to try the double-spend. There's a setting in the Android app to simply reset the blockchain. I believe it's equivalent to resetting the app to a "factory default" setting and importing the private keys. A few minutes later, the blockchain was up to date and a few minutes after that, the next block was found, confirming the transaction as a -.0001 payment to myself. Great success.