You need to edit /etc/lilo.conf/ both to define the new kernel you've just built and to identify the second ethernet card. See the installation manual and the specific naming conventions and partition numbers for your system. This is an example. Again, note that the ethernet IRQs and base addresses are specific to my system.
Your lilo.conf file has the entry for the old linux kernel, and you've already added the line to probe for the second ethernet card. You want to add an entry for your new kernel, relabel the old kernel so you can revert to it if there's a problem booting, and add an entry for your Windows partition if it isn't there.
Refer to the lilo (8) man page for details.
append="ether=11,0x6100,eth0 ether=9,0x6200,eth1 mem=128M""
boot=/dev/hda
map=/boot/map
install=/boot/boot.b
prompt
timeout=50
image=/boot/vmlinuz
label=linux
root=/dev/hda2
read-only
image=/boot/vmlinuz.original
label=original
root=/dev/hda2
read-only
other=/dev/hda1
label=win95
table=/dev/hda
Your new vmlinuz might not boot by default. The label fields in /etc/lilo.conf/ are how you specify the image to load at the boot prompt if you need to. In the example above, the image you want is labeled linux.
/sbin/lilo
sync
shutdown -r now
You should have internet access from your Linux machine after the system reboots.
If your linux kernel didn't boot by default (eg. Windows came up instead), you can redefine the default by adding the following line to /etc/rc.d/rc.local/.
/sbin/lilo -R linux
