If you copy the radial edge of the surface and flatten it out at the centre (e.g. by matching it to a horizontal line), and then revolve to create a new surface, to replace the old one, you will find that splits ok:
Note though that you will have to rebuild the entire object of which that surface is a part in order to get a closed object.
You may have other issues that are preventing a complete set of splits. A good tactic is to hide the parts you have split off so you are left with just what hasn’t separated - otherwise it is difficult to see the wood for the trees. Besides splits running through singularities look for splits that coincide with a coplanar portion of a surface. Run intersection between any recalcitrant surfaces and the splitting plane: if you don’t get a complete intersection then the surface won’t split.