How to get the angle from vertical window blinds
A hall-effect sensor, plus a small magnet attached (glue?) to the outside edge of one slat ought to be able to tell you the strength of the magnetic field, and thus the distance of the magnet from the sensor. From that, given that the rotation is with a fixed radius, you can deduce the angle while introducing zero additional friction.

Edit:
I had imagined something like:



However, looking again, this introduces difficulty in detecting the 90deg position, where the magnetic flux lines are exactly parallel to the hall-effect sensor plane.

If you experiment with this enough, you may be able to get repeatable values from the sensor to know the angle of the blind slats.

Edit 2:

The timing measure you noted in your question could also be a possibility. If you could use a microswitch or photo-interrupt sensor on the extreme ends of travel (0deg and 180deg), so that any time those extreme ends are detected, you can eliminate drift.