What are your Wi-Fi settings on your iPhone 5? Do you have it set to your 2.4 GHz band or your 5 GHz band? You said that when it does Wi-Fi it's really slow? I routinely get better speeds when using my iPhone 5 on the 2.4 GHz band than on the 5 GHz band (you'll need a dual-band router that supports both bands, such as the
http://home.cisco.com/en-apac/products/routers/E2500) . Here's a cheap one at Best Buy:
http://www.bestbuy.com/site/NETGEAR - RangeMax Dual-Band Wireless-N Router with 4 Ethernet Ports/1208844.p?id=1218234872989&skuId=1208844.
Interestingly, I get better speeds on the 5 GHz band on my iMac, but slower speeds on my iPhone 5 on the same 5 GHz band, even when both are within inches of each other.
By the way, going back to your 4S won't resolve the situation. The 4S does not support the 5 GHz band. Personally, I'm thoroughly satisfied with using my iPhone 5 on the 2.4 GHz band.