l have an old bmw 528e and the speedometer is broken. Can anyone tell me how l can fix it myself?

Well u have a few possibilities unless its a really rare never happened problem. Lets start with the easy stuff. at the rear differential there r two wires that send a signal fro the sensor in the diff is cover. Sometimes a cheap exhaust is put on & the wired get burned bad. Next lets remove the 2 bolts hat hold the sensor in. Pull it out (it can only go back in one way) if there is a lot of junk stuck to it clean it & put it back in.
Ok here is the most common problem, at its a 300 dollar part. This part can be repaired cheaply but the repair usually done not last long no matter how they do it. Ok this part is commonly called the SERVICE INDICATOR board. The name sounds wrong but its the right part. It runs all the gages & some lights & works with the trip computer if u have that common option.
To get this part out u have to remove the steering wheel
1. remove the horn button usually is the easy pop out type.
2. with a 22 MM socket & an extension plus ratchet u loosen & remove the bolt. Just let it lay in the hole. Turn the key to the on position & simply lift the wheel off.
3. Up high by the top of the clear window for the gages there r 2 small screws. remove them
4. with a butter knife lip it between the dash & gage cluster & pull it towards u till u can get a grip on the cluster. pull it a ways out. Remove all the wires. the white/ yellow & blue big plugs have a black thing on them u have to slide them back with a tiny screwdriver as a puller. then u can remove the wires.
5. once u have the cluster in ur hands set it down & remove all the screws on the outside border & 2 that r at the bottom just an inch up.
6. pull the gages away from the frame of the cluster & remove a black clip that is behind the tach (pulls straight down.
7. with a little force pull the tach away & off the board
8. pull the speedo off
9. behind the temp & fuel gage r two tiny nuts down deep in a hole & remove them.
10. (fading here my memory) l think now u can pull back the temp & gas gage with two white ribbon wires that pull off the S l board.
11. You down to the si board see the little AA looking batteries they r the problem the acid ate ur boards connections up.
12. there r 2 tiny regular slotted screws that hold the board to the white frame.
13 Replace the board
Programma has a new better board that does not have batteries. They cost retail about 300 bucks. The good news is that if ur temp gage or any other gage was dead it/they will come back to life.