There are many reports about the "defective charging jack syndrome". The charging jack (of early models?) apparently is defective, causing the radio "not to turn on" when the charger is disconnected. Looks like the switch inside the charging jack is broken and not allowing the batteries to re-connect to the radio when you remove the plug.
Another theory is that the "F1" battery fuse will open for some reason.
1) Send the transceiver back to Kenwood service, especially if it is under warranty.
2) Try to remove the battery from radio and reinstall it back.
3) You may try (AT YOUR OWN RISK!) to touch a little bit the switch inside the jack using an small screwdriver or a piece of wire.