Seriously, if doing a factory reset (hard reset) will not restore the system, I would replace the router that will not work. Since the computer/modem/first router system works, it is unlikely to have anything to do with the computer, modem, or first router settings. You have already spent hours trying to solve this, to no avail. And it would be very interesting to see if a new router #2 does the job. Good luck with this. But if you want to delve deeper, try this.
Connect each router one at a time to your computer. Check that the IP's are different for each router. Go into the Setup page for each router and be certain that they are NOT both on 192.168.0.1 or 10.0.0.1 (dependent on brand). If so, change the IP of the second router to a different IP by changing the third set of numbers, say something like 192.168.1.1 or 10.0.1.1. Just be sure that the modem IP is also different. Since the routers are in different locations, I think you can enable DHCP for both routers as long as they are not in wireless range of each other. But if in doubt, try disabling DHCP on the second router. Also set the second router to look at the first router's IP access. Again, good luck.
If you are not certain how to access the router, it is simple. Connect only one router to the computer. Call up your web browser. In the address box where you might see, for example, www.tomshardware.com , delete the current address and input the numbers that constitute the router IP, such as 192.168.0.1, and press Return. You will get a window asking for user name and password. Most routers are factory set to use 'admin' for user name, and either 'admin' or 'password' for those. Don't change 'em. Now you can look at the settings of the router. Programming a router is not hard, but it can be daunting if you have not done it before. I have to refer you to online help from whichever router manufacturer you have.