How fast of a LAN do you need for high quality mkv's?

Since you’re wired 100Mbit or Gigabit, you should be good for any h.264 .mkv file you can throw at it.

If you were playing uncompressed BluRay rips (either as .ts or .iso) I think you’d hit a wall using “Network Shares” around about the 30Mbit/sec range.