The SIM is probably held in with a little light friction. The first thing to try, because it is the least invasive, is to hold the phone firmly in one hand (low over a soft surface in case you drop it) and with the SIM slot facing downward, thump the phone against your other hand (make sure the SIM slot itself is not hitting your hand or you risk knocking the SIM right back in) - that may cause enough force to "knock" the SIM and corner of the slot out without running any risk of damage to the phone. Visibly check to see if the SIM has moved after a couple dozen taps. If it has moved at all, keep it up - this is a good patient way to get the thing out without running any risk of other damage.
Failing that, if you are careful to only grab the edges the SIM itself should be removable with a set of very fine tweezers, just grab the visible edges of the SIM and GENTLY pull it out - it should not take a great deal of force.
The broken corner of the SIM slot is the part that gets pushed up by the ejector when you insert the SIM eject pin. It's probably still located behind/under the SIM slot area, and will likely be subjected to the same friction. However, once the SIM is out, hopefully that will come out with it or by repeating step 1.