K.L.
S.,
Having had many broken bones I have vast experience bathing and keeping casts dry.
If the bath was too much movement for the finger and made it bleed you could try a shower, helpful to have a movable showerhead. One you can pull down and point in the area you need it to be in. The other option is sponge baths. One bowl of water for washiing, one for rinsing. You washe her up keeping the finger away from the water. You can then wash the fingers and palm of the hand without getting the whole thing wet.
Good luck. The good news is it won't last long! SHe'll heal and be able to take a bath again.