The speedometer on my 2001 Camry works at lower speeds but at about 60mpg it stops and doesn't move, why?

Question moved to Toyota Camry.

Number 1 tip. When changing batteries always maintain power using charger, jump leads or a couple of lengths of wire.

All modern vehicles have immobilizers. There are a number of ideas to reset or disable.

https://www.google.com/search?q=2001+toyota+camry+security+system+reset

Module location

https://www.google.com/search?q=2001+toyota+camry+security+module+location

If none of that works, you need a mobile auto-electrician.
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May be vehicle speed sensor. Any codes?

check fuse/relay

if the speedo is not sending a speed the trans will go into back up mode and the trans will not shift normally check the speed sensor

Replace your speed sensor.

electric speed sensor no side of trans one 10 mm size bolt holds in place. check fuses first, then ck speed sensor.

are you saying that the odometer does not roll over the miles and that the speed of water does register the current speed?
Usually if the odometer is not working it is a problem inside the speedometer unit in the dash.If neither is working then you may have a broken connection to the unit on the transmission that sends electrical information to the speedometer.I hope this helpsregards

The most likely cause is that the speed sensor has failed. Less likely is that the speedometer drive gear has stripped, or that the speedo meter in the dash has failed. Some details are here, around p63, but I think you would be best served by taking it to an auto electrician who has the necessary test gear

http://www.turboninjas.com/camry/be.pdf

speedometer works intermitently

It's probably not a fuse. It sounds like your vehicle speed sensor went bad in the transmission. The reason your radio stopped working was because your radio is probably equipped with speed sensitive volume and it requires a speed signal to work.