As Iain points out this looks like a job for DNS.
The iptables solution will probably be something like this, if you want to stick to it:
iptables -t nat -I PREROUTING -d 22.214.171.124 -p tcp --dport 80 -j DNAT --to 126.96.36.199:8080
iptables -t nat -I POSTROUTING -d 188.8.131.52 -j SNAT --to 184.108.40.206
The first rule does the actual redirect, the second one masks the source IP address so 220.127.116.11 returns the packets to 192.168.1.105 and the connection thus actually works. Do note that all packets will go through 192.168.1.105 at all times - it's a low level (TCP) redirect.
A nicer way to do this would be to set up a web server on 18.104.22.168 to force an HTTP redirect to 22.214.171.124:8080. This is probably much better because it doesn't mask the redirect and the users may become (probably slowly) aware of it. Also, you only have to process the initial connection on 126.96.36.199 - the rest goes directly to 188.8.131.52.
About the IPs - you aren't using 184.108.40.206/24 as a private network, right? It isn't private.