Do you need to have the mornings? Most classes are after 4pm on weekdays.
We are also in the S12 area and just recently being checking out some of the dance schools. In the end we enrolled at Bailey-Cox which does ballet and tap on Fridays at 4pm and is based in the Bannderdale Centre.
There's also June Fletcher dance school on Briarfield Road, Charnock/Gleadless, which I think is Thurday pm (best check with them) and there's Vicky Durning dance school in Dronfield on thursday's at 2:30pm. There's also Stepz dance Academy in Frenchville that does a disco class for 3 year olds on Thursdays at 4pm. Susan's Dan studio in Eckington, but can't remember the times of the classes.
Hope this helps.