First you remove engine cover.Take a pic so you can put every thing back the same as it came off. Then you should be able to see the fly wheel brake take what is left of the cable off and reinstall new cable.
If the mower has a shift knob control for "Start" "Run" and 'Stop', the kill switch will be on the end of the cable that leads to the throttle bracket. If the mower has the safety handle that shuts off the mower when it is not depressed, the kill switch will be at the end of the cable that leads from the handle to the engine. If the mower is operated with an on-off ignition switch, the kill switch wire will be spliced to the switch "off" wire some where between the throttle cable and the switch.
The kill switch is typically the ignition key. Depending on the mower it could also be a red push button labelled "emergency stop"
Broken cable?
It may not be your throttle cable it might be that your governor is stuck.
The speedo or the speedo cable might be broken and, or be dissconncted.
Broken speedometer cable.
Broken speed cable. probably doesn't have a cable it works off of a vehicle speed sensor. it could be bad
Run out of gas? check spark plugs? I would push it into the garage and call someone who knows about them.
Replace emergency brake cable Secure emergency brake pedal in its upper most position (released)
The logical problem - would be a broken speedo cable. The speedo & odometer are both run from the same cable.
BROKEN CRUISE CONTROL CABLE (CHECK AT THROTTLEBODY AND FOLLOW TO TRANSMISSION)
Certainly, any cable can stop functioning.