I open 1D -- Partner 1S -- I bid 2C (my holding was xx-Kx-AQTxx-AQxx) -- Partner bids 2D --
What do I bid, and why? (since I have to respond when partner keeps changing suit and no NT has been yet bid)
Your 2C was nonforcing; partner could have passed 2C but instead showed a preference for your first suit with 2D. This is to play. Depending on your responding style, partner could have as little as
KTxxx xxxx xx xx
or as much as
KQxx Qxx Jxx xxx
In any case, you don't have quite enough to take another bid and should pass. To bid again, you should have at least reversing strength (e.g.
Kx Kx AQTxx AQxx
which in my opinion is not quite enough to make a game-forcing jump-shift after partner responds).