You need to change destination IP on router (there where is access to this external IP), like that:
iptables -t nat -A PREROUTING -p tcp -d 59.177.X.X --dport 80 -j DNAT --to-destination 192.168.1.3:8080
Do not use REDIRECT target, it is for other purposes. If you have no access to router, therefore have no access to external IP, you can't publish your LAN servers. Then you can only talk with your provider to redirect for you one or more ports (but surely no 80), or buy your own public IP address.
P.S. Please do not use capital letters on whole post.