first step, you need to log into your linux mint via ssh, do you?
In this case you need to verify:
a. sshd is up and running on mint
you can do easily check this with the command:
sudo service sshd status
if sshd is up and running you should get something like this:
openssh-daemon (pid [nnn]) is running...
otherwise you need to install and configure that!
b. there is a firewall that block your putty request
remember, you get a timeout (not a reset, so maybe a firewall is protecting your server (the other option would be a routing problem ...)
I assume you have iptables to protect your linux box so to verify this you could:
sudo service iptables status
if you get something like iptables [xxx] is running you can simply stop it
sudo service iptables stop
and try to "putty" again ... you should get (at least) a resopnse now!
You could help us to help you specifing also the (private) ip addresses of your boxes (linux and windows and so) and may tell us about how their connected (via a router/switch/ap and so).
This are the first checks that should help you to solve this issue ... or I hope so!
Once this problem will be solved you can go further!