A bit of online research confirmed that there were likely some suitable desk chairs for Katie at Ikea. Better yet, they had cushions of the right size for our patio furniture for a good bit less than we'd found them for online. So Katie and I headed in that direction today.
We didn't end up with the chair that I had initially looked at the night before, but the one we found was eminently satisfactory. And the cushions that Ruby chewed up from the patio furniture have now been replaced with cushions that are substantially better.
I'll count this as a success any day. :)