Wifi Network - Iphone 5s grabs IP address, but doesn't complete connection

Have you tried rebooting the router by unplugging it for 30-60 seconds?

Is there an apostrophe in the name of your iPhone? If there is, rename it to something without an apostrophe in Settings > General > About > Name

Your modem is most likely handing out a 192.168.(0 or 1).n address. The 169.xxx.xxx.xxx could be anybody's.

Have someone else try to connect to your wireless. If they are successful you may be able to rule out your router. However you need to make certain that you do not have any blacklist mac addresses in your router. Way to make certain is to go into the configuration and check or just reset your router back to factory settings. Once you do that go into settings on your phone and reset all the network connections. Then try and connect.

Yes, that means your iPhone is not getting its IP address from the router. Oddly enough I just ran into this with a laptop yesterday that couldn't connect to my new router and was showing the same symptoms, which was corrected by an updated driver to the wireless card. I know that has nothing to do with this situation, but I think raymond73 has given good advice and it's definitely worth a try.

What about changing the broadcast channel for your wireless network? You can also change the options for a/b/g/n if you have that available; reading this forum, some posters have reported success changing these options along with changing the security options. Is it a dual-band router (2.5 GHz + 5 GHz?)

Well done and thanks for posting your solution!