Normally, I would suggest first that you try using the Enet port on your laptop to help narrow down the source of the problem but that's not gonna work if it's broke.

Maybe try one of those cheap USB to Ethernet adapters to get a working wired Enet port on the laptop...expensive troubleshooting if you have to buy one but maybe you know someone that's already got one laying around.
BUT...It sure sounds to me like the wireless and wired LAN connection in that modem/router are isolated from each other (I.E. not bridged) so they can't ping across between the two.
If you have the options in your modem/router config, look at the Routing Table to see if there are separate entries for the WLAN and the LAN.
Here's a few ideas from my Linksys router...details will probably be different on your modem/router.
See attached screenshot #1 for my routijng table. (I kinda doubt you'll have all the advanced features on your unit...but maybe).
My first line is the internal IP address of my network (both wireless and wired bridged together)
Second line is the external WAN (ISP) IP address.
Third line is a duplicate of Line 1...leftovers from me trying to setup FSHost on the same LAN as my FS9/FSX computer. You can ignore this one.
Fourth line is the external WAN (ISP) Gateway address.
Another one to check is the VLAN (virtual LAN) settings if you've got em.
Screenshot #2 shows my vlan page.
It shows that my wireless (the W column) is on a different vlan but it is bridged to the same vlan as wired ports 1-4.
You might also see this listed under a "Bridging Table" page or section.
Mine lists one bridge connection that is tying the wireless and wired LANs together.
I've seen a similar problem with the newer Motorola cable modem/router units. They have very few config options on the router side of the box so what we did was add a separate router (usually a Linksys) that has more config options.