Recommend me an Access Point (or router to use as an AP)

I am using a Linksys E4200V2 for my routing needs. I have 6 hardwired connections and 9 wireless connections. Due to my hardwired connections all going into one corner of my basement my router is placed there.

When I had an E3000 my wireless coverage was fairly acceptable. After a power surge took out a bunch of stuff (which I have remedied with a whole home surge protector and better surge protectors for my devices) my E4200 replacement just doesn't have as good of coverage, especially in our office. This renders wireless to our phones/tablet useless in there (wife's laptop and my desktop are hard wired when in the office). I have a fairly large home so no wireless router is going to give me great coverage in my office, which is furthest from my router and has a bunch of obstructions to go through.

I want to do an access point. I tried with an old Belkin Wireless G router, but it is only working when it wants to and there is probably a reason I stuff it in a box years ago. Of course I gave my other working Netgear N-router to my mom...so I'm out of wireless routers.

I want to buy something that:
- Is under $50
- Can do dual-band wireless N
- If it has Gigabit Ethernet ports that would be sweet as I could make use of them, but I can live without
- Can either repeat the wireless signal from my Linksys or be hard-wired as I have Ethernet connections all over my home. I'd suspect there would be a performance benefit from a hard wired unit.

I don't care if it is a router (with routing functions turned off) or a dedicated AP...I just want something that will work well.

Usage - from a hardwired perspective, I do some heavy LAN file transfers and high-bandwidth streaming on a semi-regular basis. For wireless, some random downloading (nothing time urgent), surfing and video streaming off the web is about it, and Airplay from our iOS devices (iOS stuff is only Apple stuff we have, rest is Win 7/8 and Android) with Apple TV being hardwired.