The box gutter has either not been installed correctly or as you say is too small, or both.
Are the down pipes you mentioned actually servicing the box gutter or are they for the eaves gutters?
Is there an overflow pipe on the box gutter - a small length of pipe poking out the end of the box gutter or maybe on the bottom of the gutter.
It's a bit hard to tell how much will need replacing without actually seeing it, don't suppose you could post some pictures. It sounds like water is getting into the roof space and wall cavity, this is not good as wall frames etc may be damaged.
You will almost definitly have to replace the gutter at least
As far as getting someone to fix it I could do it if I was in adelaide as I have a licence in both trades, this is not the norm and you are going to find it difficult to to find someone to do a good job in both aspects of the job.
Many tradespeople stay away from box gutters as they can be a nightmare.But try ringing a few differant blokes maybe a roofer a carpenter and a builder and when they come out really quiz them about how they are going to rectify the problem.
A couple of things to remember
1. an internal box gutter should have at least two downpipes or one downpipe and one overflow.
2. The top edge of the box gutter should extend up between the roof batten and the roof sheeting at least 150 mm.
3. If you find the person who installed the original box gutter beat him to death with a blunt instument