This is likely just a viral disease which is common in kittens and cats with poor immune systems. While there are common respiratory viruses in cats, there may be an underlying immune disease such as cat leukemia or aids that may need to be tested for. Another test if not yet performed is a fecal test for parasites since sometimes these are responsible for poor immune systems and the recovery is much longer. Some kittens just have an ongoing slow recovering process and some of them are prone to relapse. The vaccine is unlikely to make things worse but if the immune system is down then it may not work as well as it should. As long as the kitten is eating and drinking it may not need any treatment and you will have to weather the storm so to speak, however if there is a snotty or pussy nasal or eye discharge and the kitten is not eating so well, then a course of antibiotics may be needed. Speak to your vet for some guidance.
Good luck and hope the kitten does well.
Martin Slome DVM
Centre Street Animal Hospital
7700 Bathurst Street Units, 40-42
Tel. (905) 771-9855