Try HUCA (hang up, call again) and hope for a more helpful TK agent.

In my experience, results vary completely. I've been able to access my flight previously via the TK mobile app (but never via the website), though recently that (the mobile app) has failed completely. I know it has worked in the past to pick seats (this was a TK business class award booked with United miles), but now does not work (just results in a generic error). I called TK CS and an agent was reasonably helpful (let me pick seats on multiple flights, confirmed details, though she seemed a bit grumpy, impatient, and evidently was trying to get rid of me as quickly as she could).

Things go smoothly with TK until they don't, then it's the wild west. Remain flexible and keep your expectations low when it comes to customer service. Maybe things will be fine.