......can I use one of my old Airport Extremes (6th generation) to connect wirelessly to the Asus and extend the wi-fi?
Sorry, but no. Apple used a proprietary "wireless extend" technology that would only allow AirPorts to wirelessly extend the network provided by another Apple AirPort router or AirPort Time Capsule. Pretty sneaky.
Many have tried, but none have succeeded in trying to outsmart the Apple engineers.
If so, I would then connect the Airport to the TV.
IF....the AirPort Extreme could wirelessly extend the signal from the Asus router.....(it can't).....you would connect the TV to the wireless network extended by the AirPort Extreme.....or.....better yet connect the TV using an Ethernet cable to the AirPort Extreme.
Wired is always preferred to wireless.
I'm not fond of Ethernet over power line adapters, but if there is no way to run an Ethernet cable from the Asus router to the AirPort Extreme, you might consider trying a pair of Ethernet over power line adapters.
The idea here is that you already have the wiring in place.....you just need an adapter near the Asus router plugged into the wall AC socket and another adapter plugged into the wall AC socket near the AirPort Extreme. Then, connect a short Ethernet cable from the Asus router to the nearby power line adapter and do the same at the AirPort Extreme end.
Downside......There is going to some signal speed loss with power line adapters, no matter what the specifications might claim and the adapters can sometimes pick up "noise" over the AC wiring in the house.
You won't know how well......or if.....a pair of power line adapters will work in your home until you actually try them out in your home. Make sure that you understand the store's return policy before you buy if you decide to try the power line adapter option.