Dear Thomamon,
I just want to mention that .mkv is just a container format. Do you know also happen to know how the video channel was encoded? And what about the audio channel?
The reason is that possibly you only need to change the mkv container into mp4 for making it to play. In that case Re-muxing is faster. But it's only valid when the encoding is in h264 for video and AAC for audio (both supported by the MKV container). When the video and audio are not properly encoded, you need to do just that. Encoding takes more time and comes with (not necessary visible) a reduction in quality.
I use Subler as a muxer (it's free).
When you require to re-encode the video then my vote is also to use Handbrake.
That being said, when the issue is about the audio track only (=the video is in h264), which is the quite common when changing to Apple TV, I use MKV2M4V (app store) to convert the audio track an mux everything togheter in Subler.
I hope it helped.