Former Celtic star Burchill has been playing club football in Thailand while Burke has been caretaker boss since Gareth Evans left the club last month.
Burchill will combine a coaching role with playing for Livingston once his international clearance comes through.
Livingston chief executive Ged Nixon issued a statement on the club’s website expressing his delight at the acquisition.
“Following a rigorous recruitment exercise, Richie and Mark have been appointed to the position of manager and assistant manager on contracts which will run until the summer of 2015.
“Richie and Mark impressed us hugely during out recent process and we are absolutely delighted to have both on board.
“We see them as the ideal candidates to take forward our hopes and aspirations for the future – to develop our own quality young players and deliver an attractive and successful team at first team level.
“Richie has worked closely with the club since last summer when he arrived to take up a post with the Livingston Youth Foundation and he has extensive coaching experience from his time in the USA.
“Mark is at an earlier stage in his coaching career, but has all the attributes required to make a successful transition from playing into management. As his hometown club, we are delighted to give him an opportunity to start that journey.
“Most importantly, they both share our philosophy in how the game should be played, and from that perspective, they’ll build on the strong foundations we already have in place at Livingston.”
The new management team’s first game in charge is Wednesday’s home fixture against Hamilton Accies.
Livingston currently sit third in the Division One table, with 43 points from 27 games. Their chances of promotion are virtually gone however as they trail Morton by 18 points and Partick Thistle by 16.