House already wired for ethernet; how to engage?
Ah, look, my day job...

Step one - if the roof antenna is gone, and you are not otherwise using the existing cables, remove the power injector. It's just wasting power and doing nothing useful, and may be incompatible with whatever you do next. It is at minimum irrelevant, or else it is doing something and you'll notice something like video cameras stop working and revise your assessment of your system wiring.



What goes where?


Plug a patch from the Living room (LR) router into the LR wall port. 
Take a computer to    the garage and plug in a patch cable until you
find the one (there    should only be one) port that works - 
That port/cable leads to    the living room.    
Label it. (preferably, label both ends - Living room outlet is labelled with port on the patch, patch panel is labeled living room... [or a spread sheet is filled in Port number XX | Living room and then printed out and taped to the wall after it's all filled in - there is usually not room for good descriptive labelling on the panel itself.]


Now, plug another patch cable into the living room port (in the garage patch panel) leaving the living room port in the living room connected to the router.


Plug it into any other port in the garage patch panel. 
Go back in the    house and plug a computer into wall jacks until you
find one (there should only be one) that  now  works. 
Label it (preferable method is as above.)


Repeat until done. If there are jacks elsewhere than in the house, include those. Presumably you can see the wire that leads to the roof and know you don't need to go up there with a computer to find it, and probably don't want to connect it at all. If you have checked every jack you know of and cannot find one that works, note that at the patch panel and move on (there's either some you don't know, or wires are broken/damaged.)



With that done:

You'll probably want a switch in the garage, unless activating only one of the other ports in the house will meet your actual need - that would cost the least. 

The best guess I can make from counting wires in the too-small picture is that you have 13 coming in there rather than the 7 your diagram admits to - if you want all of those to work your minimum switch size would be 16. If you only want some of them to work, a 4 or 8 port switch would be less expensive, typically. 

I cannot make out from the small picture what obscenity has been done with what appears to be a rack mount patch panel in the garage that's either hanging from the wires (bad, bad, bad idea) or screwed to the wall in what would appear to be a direction that would make the patch ports inaccessible, normally. You can get a small wall mount rack to clean that up rather inexpensively, or even do something creative with blocks of wood and screws that would be better than what little I can infer from the picture. 

At that point, your path forward forks a lot depending on what you want and what you have. Personally, I hate having the crappy thing from the cable company doing whole-house WiFi wherever the cable happens to terminate, so I'd either turn that off or turn its power way down and put other WiFI access points (not routers) elsewhere in the house for better distributed signal. With the fancy high speed wireless that runs on 5 GHz, if you can't see the access point from where you are working with your device, your signal will be badly degraded or your device will switch to less fancy and slower 2.4 GHz, since that penetrates walls far more effectively. You may not care, you may be happy with the WiFi (but then why are you even looking at the cables? I suspect you are not totally happy with your WiFi) or you may just want wired connections throughout the house - figure out what you have and what you want and pointing you forward without making assumptions becomes a lot easier.

For instance, it may be a bad assumption that there is Cable-TV coax run to the garage, since the previous method might have been the antenna on the roof being the source of the internet in the house. 

Somewhere towards the end of the process, using either screw-attach cable ties or double-sided velcro screwed to the wall (easier to reconfigure without having to cut and throw away all the cable ties) to tidy up that mess of cables would fall under "...in a workman like manner" i.e. not looking like total crud.