After first making sure you have no warranty, I would crack open the case and carefully check the connection points for the ribbon cables. Check beyond the normal warranty. Some credit cards had considerable "extended" coverage. Your alternatives are few. You can buy a new unit ($80 on ebay +$15 shipping) or live with the DSLR screen to provide the functionality. $100 on Amazon with free shipping.
If you decide to crack open the case, I would removed the batteries, being careful of the door as that seems to be a weak link with these units. Then look around in the battery compartment for some place to wedge in a wide blade screwdriver to hopefully split the case. The two halves of the case are likely held together by tabs that snap into a slot. Screws might be used but nothing is apparent in web pictures. Feel around for screws on the device, especially in areas that might be hidden by a label or something similar.
Similar Yongnuo units show a screen that has 3 or 4 connection points. You would need to insert some material over the connectors to maintain pressure when the device is assembled. You would not want it too thick. I'd start with electrical tape, cut to cover the connector and maybe about 6 layers thick. You also have to stay away from what are called capacitors as they can retain a high voltage. Work with some thick rubber gloves.
Edit: I did find a web picture that shows the mounting shoe with four screws. You might loosen them and see if the case separates.
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