Let's divide the problem into smaller pieces - is this an N7 LTE problem or ASUS router problem?
Get another router (I use the Asus RT-N66U) and test if it works with the N7 LTE. If it does, then it's your' router.
If it does NOT work wit hth eN7 LTE, then it's your N7 LTE probably have a bad chip.
AT&T Galaxy S4 Active with 4G LTE
Stock ROM - rooted
Official AT&T Jelly Bean 4.2
Nexus 7 (2013)
Stock[/B] ROM - rooted
Official Jelly Bean 4.3
Galaxy Note 8.0
Google Nexus 7 Tablet - Official Google JellyBean 4.2
AT&T Galaxy Note with 4G LTE - Official AT&T Ice Cream Sandwich 4.0.4