Removing it yourself depends on if marking up the aluminum case is acceptable each side of the broken screw.

If you can slot it with a Dremel abrasive disk, use the non-reinforced type that are about 3/4" that is dressed down with the dressing stick thats included in the bit assortment with the Dremel. Making the disk small as possible to still do the job will reduce the amount of marking that will happen on the case.

SECURE the case somehow and bring the disk down nice and steady with both hands on top the bolt to form a slot for a flat screwdriver.

Wash and pre-heat the case around 250 degrees to loosen any old castor or loctite that might be in the threads and give it a try. Sometimes the heat and vibration that forms when you grind the slot can bypass this step, so try it LIGHTLY with your screwdriver and if it doesn't tell you its moving..stop before you strip out your slot and have to go deeper.

If you bring it to a machine shop, they will have to have a carbide end mill small enough to be below the minor diameter of the threads, and also be able to clamp the block to the table of the mill by its mounting legs...NOT squeezed in a vise, and be able to locate well over the hole, use lots of air, and a slow hand feed up with the table.

A tool and die shop will have one step better and be able to burn out the screw using an EDM machine, and the ability of the operator will be higher, and those boys will be more experienced than a job shop for removing broken screws.

If you find a tool and die shop in your area and you take the time to drive over, try and explain to the guy you're just a hobbiest and not a company, and he'll probably give you a break on the price. Shop time isn't cheap in a tool and die shop, but this usually doesn't take much time for them to do, and even tell them there's no hurry on the job and they'll fit it in at the end of the day or such and just take another drive over to pick it up.

Good luck keeping chips out of the bearings and being able to wash any out when your done. I've never had any luck flushing bearings clean inside the case well enough to trust running them again, even taking extra precautions not to get any chips or grinding grit inside them. Keep a new set of bearings in mind before attempting this.

Broken screws can be a bummer, and takes more time and money than you think by just looking at it sometimes. Nothing is worse than looking at a simple broken screw inside a hole, and then finding out the mountain of work it takes to fix it.

I always hope the Dremel method works and is acceptable before any of the other options have to be considered. Also consider what the other guys here are saying too, all good advice.

If you're attempting to do any kind of free-hand drilling or eye-balling the center to get it out...I suggest working on getting the top of the screw flat and a starter hole centered in it with a dremel bit...then use the drill. A properly prepped starter hole for the drill bit means the whole world on how sucessful you'll be on not damaging the case threads.