My old macbook air would not turn back on.

Try an SMC Reset if you haven't already.

Unplug the charger from the electrical outlet for a minute to reset the charger's internal circuitry.

Since you mentioned "old" MBAir I'm assuming it uses a Magsafe charger. Make sure there are no staples or other small pieces of metal stuck within the Magsafe port on the laptop which will prevent the charger from seating correctly.

Here is an Apple article for troubleshooting Magsafe charging issues:

https://support.apple.com/en-us/HT203207

If the battery was completely drained for any amount of time it can prevent the laptop from powering on at all or even charging. If the Magsafe LED is lit allow the laptop to charge for a day or so. Otherwise if a completely drained battery is actually the issue, then it may require disconnecting the battery and powering on the laptop using just the charger followed by reconnecting the battery while the laptop is still on. Leave the battery charge at least 10% before powering off or disconnecting the charger or permanent damage to the battery could occur. Extreme care must be taken not to have any metal or conductive objects/surfaces come in contact with the exposed internal components.

Otherwise you will need to have Apple or an Apple Authorized Service Provider examine the laptop to provide an estimate for repair.

press the power button down for 10 seconds

this occasionally happens to my spare 2010 MacBook air

or

you could use the old logic board start up technique

my other MacBook power button does not work but i can still power that on.

you would need to search how to do this online.