Questions:
1. What is the make and model of Linksys router?
2. What does your network topology look like?
3. If the Linksys router has the capability of being a DHCP server and NAT, why are you bothering to run ICS?
If the router is really the device I think it is (a true NAT/Firewall box), then you're doing overkill by running ICS on a Windows box; the router does that job just fine. And by the way, ICS running on most versions of Windows is a completely POS and nothing but trouble most of the time.
Anyway, if the Linksys router is really one of those NAT/Fireall boxes, then it should have a web browser interface that allows you to set its configuration and such. Not sure if it has a screen that'll show DHCP leases or not. Maybe we can look that up once we know which one it is.
/me goes off to see if ICS has a "leases" screen or not... Might need to know which version of ICS on which Windows OS you're running...