Novice router user

I recently bought the Amazon Fire TV Stick and am having problems with it maintaining a connection with my cheap, 4-yr.-old D-Link DIR-601 router. The connections are effemeral, in that they work much of the time, but randomly refuse to connect or tell me my connection is too slow. So I'm considering buying a MUCH faster router, like an ASUS RT-AC68U or -68W (the latter being noticeably cheaper). The only devices in my home using wireless technology are my laptop, my wife's laptop, my Canon printer, and the Fire stick.

Given the above, I have some questions:

1. Which of these routers will give me the performance I need for the Firestick?
2. Is buying one of these (expensive) routers overkill?
3. Do you have any cheaper recommendations for solving my problem?

Any help is much appreciated.