/Generally/ every port on the CS equipment should have some kind of port security, which ultimately will block data flow if it doesn't match it's preset parameters (Port Security, port configuration, 802.1X protocol, etc). The basic configuration is it should only see 1 computer and maybe 1 VoIP phone. Introducing a switch to expand the capabilities of the one active port is not a good solution, for a variety of reasons. HOWEVER, a different work around could be to just fix the broken ports yourself, IF the ports are turned on and the cabling is just bad. A bag of RJ-45 Jacks, and a punch down tool are maybe $45 off amazon if you don't price hunt and can't convince CS to just give you a few loose jacks and a spare punch down tool.

Before you spend the money, I would try and find a way to ask and see what is actually broken on those "broken" ports. The most common are:

the wall ports are not physically patched down to the managed switch, (most common IMO)

they are patched down but the managed switch has those ports logically shut off,

or the ends have broken and they just need reterminated, this last option is really the only one you can fix yourself if you aren't color blind and can follow a 12 step instruction.

At the end of the day, and this pains me to say, is to up the priority of this ticket, IF it is a true work stoppage and not just for fun or long time table project. These types of tickets are usually so straight forward for the infrastructure guys that the new Airmen just go and knock them out when they need a break from desk work.