My son is 15 months (almost) and is still in a "baby bath tub." We got the Primo Eurobath from Babies R Us and its supposed to work until the baby is 2 years, depending on the kid's size. He can lay down on one side and sit up on the other. He only sits up now. The thing is HUGE, so if your living situation won't allow for an extra thing, there are others that might work.
Also, do you have a tub mat on the bottom for extra grip?
If you want to stick to your tub, I would take a bath with your daughter until she shows signs of playfulness and wanting to explore the bath beyond your proximity. Think of yourself. Did you easily transition to anything with just one try? The sink is very different from the tub. Try to put yourself in her shoes.
When we travel, we don't have the luxury of a toddler/baby tub. Our husband takes a bath with him, straddling our son between his legs. He thinks of it as extra bonding time.
It may be helpful to get a space heater, since you won't be fully submerged as if you were taking an "adult" bath.