ocean13, best you do is contact an immigration lawyer as all you will get here in this forum isn't really rock solid. I consider it highly unlikely however that you're illegally here. Most countries admit you for business / leisure reasons with the same status as long as you a) stated correctly what you are going to do and b) are not employed by a local company. This is a key difference. Example - you come here to train some staff of some company. This company pays your UK company which pays you - no problem. If you're a freelancer, though, and write the invoice yourself - then it is a problem. But no, the thousands of business folks occupying the Hyatts and Hiltons aren't illegal immigrants.
There is usually a maximum stay set for such business trips, though I'd be surprised if it was less than half a year. So shouldn't be a problem for you. Make sure though you don't get a penny from your local office until you have a work visa.
Anyway, all I wrote here is generic and applies to what I know from former colleagues who worked in the US, in Singapore or (15 years ago) my dad in Canada. You'd better quickly sort out the specifics with a local immigration lawyer.
Since your presence here seems to be of some importance to your company, I'd get that lawyer asap and have him sort out with your current agent what the problems are. I've no idea how long a work visa takes (my student visa took three days
) but if it takes three month they should be able to name a problem.