I recommend "Goo Gone" cleaner, which is a solvent that's worked great for me in the past with similar situations. It's sold at hardware stores, near all the other typical cleaning chemicals that are usually all in one place at the store. Goo Gone is in a drip bottle, not a spray bottle, so it's easy to target the areas near the edge of the disc. Work your way slowly to the center of the disc, after attacking the edges first, to avoid ripping up your surface by trying to yank the disc up too fast.
I put the black disc on my car dashboard. It seems to work great. The TomTom instructions, on how to install the mounting hardware, were hieroglyphs! It took me many tries to figure it out. As frankxs said, the instructions are often misleading and downright wrong.
I use the suction cup mounting bracket to stick to the black disc, which as been glued to my dashboard. This is much better than using the suction cup to stick directly to the windshield. The weight of the TomTom unit itself will pull it down. It is rather distracting, thus dangerous, to have the TomTom fall off while you are driving!
By using the disc on the dashboard, and sticking the suction cup to that, I'm using the suction to point down instead of up. This means that gravity helps my TomTom stick better, instead of working against gravity by trying to pull it down off the windshield.
That said, I would love a professional mounting kit for the TomTom, if something like that would ever be released! Car stereo installers could use it to mount the TomTom in such a way that all the wires would go underneath the dashboard, and the mount would be rock solid, real nuts and bolts, no suction or glue. Does such a thing already exist?