The greeks had two institutions we could use: ostracism and lottery. We should be able to vote on those we want out of politics and the most voted would have to seat it out. And a small portion of representatives should be chosen by lottery. Maybe 5% or 10%. These representatives, obviously, would not have to answer to their sponsors.
I don't think representatives need to have any particular competencies beyond being representative of the interests of the population. Hired staff can take care of things that require particular abilities.
I reckon that a wholly random set of representatives would act like a direct democracy where every voter is free to fully focus on the issues, is allowed an entire staff to help them, and can consult and coordinate effectively with other voters. I think that could work pretty well.