IMHO it would be a good idea to use a USB drive for backup and tce storage. This way you can save your network settings and any extensions that you may use. Use these steps (since you do not have a network connection):
1. Create the directory ./tce/optional in the root of the usb drive. (ie mkdir -p /mnt/sda1/tce/optional)
2. Download and copy the firmware-2.6.29.1-tinycore.tcz extension and md5.txt to the ./tce/optional directory on the usb drive.
3. Edit the boot line and add the bootcodes: tce=sda1 backup=sda1 waitusb=5
4. Boot the computer with the usb drive plugged into a working usb port.
5. Use the appbrowser to manually load the firmware extension, connect your network card, then use appbrowser to download any other desired extensions.
6. Use the Apps_Audit tool in the Control Panel to add the firmware and other extensions to the onboot list.
Notes: this assumes the computer will add the usb drive as sda1, it may be added as a different device depending on the board.