Hi. My name is*****'m a camera repair technician with over 20 years in the business and I'll be glad to help you with your question today.
This is most likely the infamous Nikon flash switch in the hinge mechanism of the flash head. This usually requires removing th top cover, which means you need to disassemble the rest of it first. Not a job you want to try at home. For me, it's a simple repair. Every Nikon that uses this kind of switch will need to be repaired at one time or another. Leave it to the great mind of Nikon to build cameras with built-in flaws. Unfortunately, this is going to take a technician to repair. Trust me, I've done hundreds of them and there's soldering involved. I'll be glad to recommend a good shop for you if you'll tell me where you're located.
When the flash head goes up, does the top LCD display switch to the sync speed?
This further indicates the flash switch is in need of repair. As I stated earlier, this switch is located in the flash head hinge mechanism and will need to be repair by a technician familiar with the camera design. There is 300 volts running through the flash circuit which, if shorted out, can destroy the camera or at the vry least, give you a nasty shock.
Just a note to see if I can be of any further assistance?