How to sync controllers when usb doesn't work?

There are a few things you can try, but eventually you'll have to probably take the console in for repair since it seems the USB Interface IC is probably toast. Repairing it on your own is possible but it's hard work and you can damage the circuit (more about this later).

First thing I would try is get a different USB cable and try all ports again. It's quite possible the cable failed but that's very unlikely.

You can use a battery pack or micro USB charger (most cellphones have one of those) to charge the controller but since you said your controller desynced, that won't help you much.

Next, you can buy or borrow a second controller and use it to sync the other one.

Lastly, if different cables fail and the second controller isn't working via USB, you need to face the fact that the USB Interface IC failed.

Here's a video on how to repair it:

I absolutely don't recommend trying this unless you have previous experience.

You cannot use an HDMI to USB adapter to control the PS4.