OnTheAsile is right about the WalMart straps. They're strong, easy to use, cheap, and readilly available anywhere in the US.

However, I've never been very confident in Velcro as a holding fastener. It's great for keeping flaps and pockets closed, but it doesn't hold much weight unless you apply it in fairly large quantities, and it's not great in an environment where you might have clothing - it collects fuzz, lint, and loose threads, not to mention that it can cause frizz on some sweathers.

There might be a better way to attach the strap to the bag, with simple techniques, but I'm not at all familiar with Tumi bags. Can you post some more detailed pics of the inside of the compartment? Maybe I can figure something out just by looking at it. I've modified several of my own bags and suitcases with additional straps and other stuff.