How to get broken screw out of wall anchor

There is quite an 'overhang' of the screw (24 mm) from the wall. That's why you need at least 6 mm screws in diameter. It's head is about the maximum you have.

That implies 12 mm anchors, which are 60 mm long here.
The resulting screw would be 6x80, or 6x90, if you have longer anchors.

Remarks:

• the overhang should be as short as possible, that's why you shouldn't push the anchor unnecessarily deep into the wall. Too big lever would strain the screw, the anchor and the wall.
• I'd drill the hole not horizontally, but several degrees (5° - 10° from horizontal line) from top to bottom. It wouldn't let the screw to be pulled out when it settles (lowers its head) a bit.

There is a great explanation of hole sizes here: http://l-36.com/woodscrewpilot.php. Some key points:

• You can easily find the actual dimensions by screw size in size charts with a google search, such as http://www.raygirling.com/screwmes.htm. Your #8 screw has threads with a diameter of approximately 5/32" (0.164"), and a root diameter of approximately 7/64" (0.112"). The root refers to the "inner diameter", the solid shaft inside the threads.

• Ignoring the countersink, you may need a pilot hole or a clearance hole. The clearance hole is slightly larger than the thread diameter so the screw can slip in and out. When you screw two pieces together, if the threads grab in both pieces, they can hold the pieces apart. A clearance hole is used in the first piece so that the threads do work only in the second piece and can pull the pieces together.

  The threads are shaped like a wedge and make space for themselves by compressing the wood. But there are limits to how much the wood can be compressed. The pilot hole allows space for the solid "core" of the screw so that the screw doesn't split the wood. The clearance hole size isn't critical but the pilot hole size is.

• The are many guidelines for pilot hole size, and the recommended size differs for hardwood vs. softwood (hardwood splits more easily). There are sometimes separate recommendations for brass screws, such as yours, because the metal is soft. Some sources recommend using a hair larger hole for brass, others recommend just inserting and removing a steel screw first to do the heavy work of cutting the threads and compressing the wood around the screw.

• Most pilot hole guides are based on achieving maximum strength as determined by the American Wood Council (AWC). Their recommendation is for 70% of root diameter for softwood and 90% of root diameter for hardwood. The discussion I linked to describes that as unnecessarily risking splitting, especially near an edge, in order to gain a trivial increase in strength. It also makes inserting the screw much more difficult.

• If you are using brass screws, the application is likely more decorative than structural and the last iota of strength is probably a secondary consideration. The recommendations of the author I cited are generally based on 85% of root diameter for softwood and just the next drill size larger than the root diameter for hardwood (which ranges from 102% to 116% of root diameter). Those values should be good for brass screws. Some other tables recommend 100% of root diameter for hardwood, which should also be fine.

  So the recommended pilot hole for your #8 brass screw would be 3/32 for softwood (vs. AWS value of 5/64), and 1/8 or 7/64 for hardwood (vs. AWS value of 3/32).