I know its a frustrating time trying to get through to anybody, best thing I can say is to call at 8am on the dot, on 0345 454 111 choosing options 1-1-4-4 to leave and they should be able to sort you out.
It's an absolute disgrace how difficult Virgin make it to cancel your account. I run an online Subscription business and my card processors insist that I have an online option to cancel any recurring bills. Why aren't Virgin bound by the same rules? It's a joke, cancelling should be a simple process.
If you are trying to ring after work then yes, it will be difficult and could take several hours to get through to someone. That's why you don't bother and just send a letter instead, 10 minutes max. Easy. Or ring first thing in the morning when queue times are low.