You don't need to blast anywhere, just a casual drive around the block for 5-10 minutes will provide enough time for the ECU to reset the faults.
However, to reset it the TMPS, you have to press and hold the TPMS button until the DIS beeps. Just make sure all your tires are at the correct pressure before you do.
There are two versions of TPMS for the Mk2; direct and indirect. North American TTs have direct with tire pressure sensors built into the valve stems. European are indirect using a sensor to detect change in wheel diameter based on rotation speed to determine if there's a loss of pressure. For...
www.ttforum.co.uk
You can find an owners manual
here. It might not be for your location or year, but it should at least point you in the right direction.