Oh, OP, I've been where you are, and it's absolutely awful. 
My tips would be to be as kind to yourself as possible. Go as easy as you can on stuff you 'have' to do. Make as much time as you can for doing things that are pleasurable, or usually pleasurable, even if you can't feel that pleasure right now. It will come back.
Try to step back from any feelings that you must be 'productive' at all times. The more time you can engage in pleasure, for pleasure's sake, and no other reason, the better. Make time for new and enriching experiences. If you like art, try and find new, interesting exhibitions to visit. If you like country walks, check out some new ones and/or revisit old favourites. Get out into nature as much as you can.
Eat well. Sleep like a king.
The MBSR suggestion is a good one. I would suggest also to consider trying hypnosis. You could see a hypnotherapist, or you might find listening to some downloads would help. Look for downloads targeted at relaxation or joy/pleasure. It can be very powerful.
Lastly, if you have any spiritual leanings, now is the time to explore them.
Burnout is a sign we have been doing something wrong. Often, that we have been ploughing the wrong furrow, too hard. It can be an opportunity to re-orient life for the better.
Hope you're feeling better very soon. But give yourself time.