This is the type of USB connected device you would use to add dual band wireless connectivity to a laptop: WNDA4100
There is no need for wires.
As I stated previously, the antennae on routers and wireless adapters only receive the radio signal through the air. The frequency of that signal, 2.4GHz or 5GHz, is generated inside the router. The antennae do not determine the frequency.
The "PCI-e wireless card" you show in that picture is actually the laptop's "Wireless Adapter". Yes it is a card, but it is an integral part of the laptop. That adapter must be a dual band adapter in order to receive signals on both bands. Adding more antennae wires to it will not change it from a single band adapter to a dual band adapter. That can only be done on the adapter's circuit board.
You may be able to get a replacement adapter for that laptop that is dual band, but don't count on every manufacturer having one.
bobafetthotmail's idea of talking to the IT guys at your school is a very good one.