Engine will not turn over

13 volts isn't a problem as a standard auto battery will only put out 12.6 volts with a full state of charge. Anything above that is from the charging system. From the description I would check the cables at both ends as rust or corrosion could easily cause this problem. If you have a voltmeter you could also do a test on the battery by connecting it to the battery terminals, then try to start the engine. If the voltage stays the same when you hear the click you have a bad connection. The starter will pull a large amount of current and while starting the battery voltage can easily drop to 10 volts or so. Now if you see the large voltage drop but still only hear the click and no starter motion it is more likely a bad starting motor.

What does it do when you turn the key? Just click or does it do nothing? If you turn on the lights and try to start it what do the lights do? How about if you turn the key on and turn the blower motor up high does it run at full speed?
To do some testing beyond that you will need a simple test light and a couple short pieces of wire.
https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

As you have power inside and said it hummed or clicked earlier I think the fuse link is okay. If it clicks when you turn the key to start the other fuse in the passenger compartment fuse box should be okay as well. The next step will be to test the relay to verify if it is working or not. To do that you will remove it from the box and use the test light and one piece of wire. This will also test if the issue is the starter motor or in the vehicles internal wiring so it makes the process faster. With the relay out and looking into the socket it came out of you will have 4 terminals. Three will be facing one direction while the last will be 90 degrees different like the blue cube below. With the key on and the test light connected to a ground like the battery negative post, use the probe to check the 4 terminals. Either the single cross or the terminal directly across from it should have battery power. At this point we will do a jumper test to see if the problem is the relay, the starter or in the rest of the control circuitry. Take a piece of wire. Slip it into the terminal that has battery power. Now just touch it to the terminal directly across from it like the line in the image. If the starter and it's circuitry is good it should spin the engine over. If nothing happens but a click the starter is the likely issue.
If the engine turns over as it should we have more testing. The next step is to remove that jumper wire. Now take the test light and check the outer two terminals. With the test light connected to battery positive the light should come on bright on one terminal which is the ground side. Then connect the light back to ground and have someone hold the key in the start position, Transmission in park. The opposite terminal should now have power if it does, the relay is bad. If it doesn't then have the person hold the brake and shift the car into neutral, does the light come on now? If yes the neutral switch for the transmission is failing. If still no light then put it back into park, put the relay back in and we will test some more.
The next item would be if your car has the optional anti-theft box in it. That has a relay that is behind the left side of the dash. It is the black box item in the 5th image. It may have failed internally. To test if it is the issue you need to look at the wiring on it for a White wire and a White wire with a black stripe. Those wires are the power from the ignition switch and out to the transmission switch. If you unplug that relay and put a jumper between the two harness side terminals for those two wires you will bypass that relay and try to start the engine, if it not turns over, that relay or the alarm system has a failure. Now if you have tested all of this and it did turn over when you jumped the starter relay at the beginning but the key still isn't working, we need to test the transmission safety switch that is on the transmission itself. On it there will be a wiring harness with ten pins in it. For this testing you will leave the transmission in park, then unplug the wire connection. If you look at the last two images you can see the terminal layout. Pins 9 and 10 are the ones you need. If you take the test light and connect it to ground and with the key held in the start position you touch pin 10 in the harness connector it should light up. If it does, use the short jumper to connect it to pin 9 next to it. That will bypass the switch. If it is failing the engine should turn over when the key is turned to start. I'm hoping you discover the issue before this however.

Replacing the relays is as simple as it sounds, unplug the old plug in the new.
In the case of the anti-theft relay it could be the system itself, but it is possible to test it as well. The neutral switch isn't difficult to replace but I'll wait until you do some testing before going to the other systems.
Please let us know what you find.