Wow, very broad question. Depends on what you want to do, and relatedly what season this would be.

We've traveled with a kid about the same age to both Japan and Taiwan.

By the date of your posting and child's age I'm assuming your trip will be between December and March. This will be pretty cold in Japan, even in Okinawa. We have been at a beach resort in Okinawa in December and it's not pleasant. You have to bundle up. And if you go to a ski resort, even if it's a club med with a kids club, your daughter won't be able to partake too much as she'll be barely able to walk, much less ski. The normal kids club at club meds start at age 4. I've taken my son to Hokkaido when he was 3 and that was a total bust -- basically it was a chore just to get him into boots and I had to carry him down. Same for Taiwan beaches. Kenting will be too cold, weather-wise, for the beach. The only place I can think of in Taiwan that will be decent is the Silks Hotel at Taroko gorge. Luxurious (for Taiwan) but the main attraction is the views. If you want to go on hikes though you'll probably need to carry your daughter or push her in a stroller. In any case I think 7-10 days is too long for any of these options.

If if were me, I would go to Hawaii. Or maybe Southeast Asia. Plenty of nice resorts for a little kid.