I had a similar experience yesterday. I played against someone (don't know if it is appropriate to name them) who would simply type "faster" during my turns, in an attempt to get me to hurry up. After a few turns this person simply typed "ok", and proceeded to play each of their turns with about 30 seconds between playing each card, and treasures individually. I didn't want to resign because this guy was about 3 levels below me, and I had a good chance of winning (and did win). The game went on for about 30 minutes, during which time I applied for a credit card.
There is a feature to force an opponent to resign if they are taking too long on an individual action, e.g. if they have just disappeared. But there is nothing in place to prevent this system being abused by players who are happy to deliberately drag their turns on one click at a time. You could even imagine that, because of the levelling system, you could increase your level artificially if you pulled this stunt on higher level players repeatedly and got them to resign out of frustration.
One solution would be a time limit that wasn't for a single action, as it is now, but instead a time limit per turn, or a time limit within which you have to play all your turns (I think there is a chess system like this...). A time limit per turn could still be abused, but if you only had say 10 minutes to play all your turns, then there is no chance that you could just drag it out almost indefinitely. Ultimately, it would still be down to your opponent if they want to make you resign at that stage.