We have what we call 'the non compost heap' It is made up of the things like yours, weed roots and so on which you are not supposed to put on the proper heap. It has been covered for the last 4 years with black plastic which keeps it looking neat. Lifting the plastic the other day , the stuff has rotted down beautifully. Now it may well be that the seed weeds are still in it, but I intend using it soon on the Veg patches where the soil has become depleted over the years.
So, I would suggest that you cover it and wait another couple of years, then use it as compost and take the risk of weed contamination.

I like owdboggy's idea but an alternative would be to fork out all the weeds/roots etc and burn them. Cover the remains with good "fresh" manure and the heat from that should kill off the seeds.

I was going to say similar to Ian. Heat should kill off most seeds and get rid of the roots/woody bits in your green bin this will help speed up the rotting down process. As well as plastic you could cover it with old carpet (or try getting some free from 'Freecycle') over the winter to help keep it warm

Letting the roots dry thoroughly should also kill most of them.