My PC doesn't support miracast. Will android TV allow me to mirror my full desktop like a chromecast and let me watch hardware accelerated MKV videos from daum player smoothly?
Also, I'm assuming it will run Kodi smoothly at 1080p?
Perhaps not the best place to ask this question, but are there any other streaming technology that would serve those two use cases?
AndroidTV is the operating system, not the device.
So yes, it's capable of any and everything
It's also capable of falling at all of the items you're asking about.
Desktop mirroring via Google cast is beta and from my experience, not ideal for low latency high framerate stuff.
Install the kodi app onto your Android TV from the play store. It can 100% handle your use case
All devices running the Android TV platform have the Google Cast capability. So if you have any experience with Chromecasts, it's the same exact thing on different hardware.
The Shield TV can transcode multiple 1080p streams and direct stream 4K. It can easily handle 1080p in Kodi. The cheaper devices like the Nexus Player and upcoming Xiaomi Mi Box will definitely handle 1080p. If you've got some funky kind of 10-bit blu-ray rip in some obscure codec, then you might run into problems. The resolution is not going to cause a problem though.
If you want Kodi and mirroring from PC, then an Android TV device sounds about right. To replace the mirroring functionality of Google Cast, you'd need either wireless HDMI, DLNA, or Miracast support on your PC. For Kodi, damn near anything will run Kodi. If you had a Mac, then AppleTV would be the best option for mirroring.
Also, I don't know what "daum player" is but if you're casting local media then I would recommend using Kodi or Plex for a huge library, or casting with Videostream to just browse for files in your file system.