There are a meriod of possible reasons why your iPhone 5 is not seeing your Wi-Fi Network. While it could be one of several Router issues, it also could be your iPhone 5 harware or software. There are a number of things you could try on your iPhone without changing your Router settings. Firstly though, I am wondering if you have other Devices that are currently still working with your Router / Network.
If the answer is Yes, other devices are accessing Wi-Fi just fine, then we have to asume that your iPhone is currently exhibiting an incompatability. My guess is that it is Software rather than Hardware. As such, I think it would be premature to run down to the Apple Store and exchange your iPhone.
I guess the first thing I would do is go to Setting > General > Cell and turn Data to "Off". Then go back into Settings > Wi-Fi and try to see if the Network is now discovered. If yes, select it and see if you can Join. Don't give up too soon, as I have found that in certain Wi-Fi Networks it can take the iPhone 5 up to 15 or 20 minutes to make the connection once the Network is in-fact "Discovered". If this doesn't allow your iPhone to "Discover" your Network then go to Setting > General > Reset (bottom of page), and select Network (Reset). Now go back and see if Setting > Wi-Fi discovers the Network and if you can join.
If that doesn't work then you may need to try one more thing prior to getting Apple Support involved. Try Resetting the iPhone 5 to Factory Settings. This will require that you connect to iTunes and Backup your iPhone's contents. To do this on your computer you will need to go into your iPhone and turn iCloud Backup to "Off" first. You will also need to transfer any "Purchasers" from your iPhone to your iTunes first. Now you can do a full Backup and after that a Reset. When asked if you want to Apply and Setting Changes that you may have put on your iPhone Say "No". What you will have after the Reset is an Opportunity to Setup the iPhone from Scratch as a new out-of-the-box iPhone 5. Follow the screen-led instruction to do this Setup. Don't worry about your former Content (Apps, Music, Photos, etc.) as you can Sync back to all that stuff latter.
Now take your New iPhone and try to connect to your Wi-Fi. The Setup procedures will specifically provide a Pop-Up that gives you the chance to Connect to your Wi-Fi.
If none of this works make an appointment by for an AppleCare Call or Store Visit. Good Luck!