I am (was) also a nanny.
although not normally, i too have had a shower at most of the places i have worked.
Most of the houses i have worked in have had a swimming pool.... so therefore after swimming me and the children all want a shower at the same time. (just to add children older and perfectly capable of showering themselevss too)
I have never used the parents en suite though after swimming, but have used the guest en suite.
When proxy parenting my ex ex charges i always stayed in my bosses room and used their en suite, but i cant imagine anyone wold have issues with that (as thats where my bosses wanted me to stay etc, and know one can do a week without a bath / shower!)
In the summer when weraing crocs id always wash my feet in the bath with the children (they got filthy!) (only washed feet, didnt get in and bath with them or anything).
When having my bathroom 'done' I asked my bosses who were more than happy to let me have a bath shower there, while the work took place. (when they hd their kitchen done and had a few days without a washing machine i took their washing back to my house... swings and roundabouts and all that)
Id NEVER even think to take my own towel, and would just use a clean one out of the airing cupboard, and add it to the wash basket to wash later.
I see no reason to disbelieve your daughter, and there may be a normal explanation. Yes a bit odd but hardly gross misconduct either. (though i dont think she needs to use your bathroom, unless perhaps you only have a bath in childrens bathroom, and the shower is in the en suite?)
I think you should ask her about it,... if she has a rational explanation... umm she was opening some custard which squirted all over her hair? then ask she doesnt use YOUR en suite, if its more like her shower is broken, make it quite clear that you expect at the very least to be asked if its ok.
Also say if she ever needs to wash herself (say child vomits all over her) can she please use the family bathroom rather than yours!
That is not an unreasonable request IMO