Until very recently I spent years working for the housing department within a council.
5 years in temporary accommodation, within London, is not considered an exceptionally long time. Many people are eventually offered permanent accommodation outside of london (sometimes miles and miles away), due to the simple fact that there are SO FEW properties available. If you want to stay in London you are likely to have a very long wait.
With regards to your neighbour - there will have likely been circumstances within her situation that you are not aware of, that placed her as higher priority than you. Councils and housing associations are heavily governed by external bodies, ensuring that housing policies are adhered to.
In the different housing organisations I have worked in, pest control has always been tenants responsibility - unless the infestation is in communal areas of a block of flats etc.
You have been given some great advice up-thread in how to treat and prevent infestations (if they are confined to inside your home only).
Low income families can often receive highly reduced rates of pest control is needed to come out (in the event of an infestation within the property).
Many homes do get mould - you can use mould detergent to clean. How large is the affected area? Where abouts is the mould?
Keep a window or two slightly open when you can, or vents permanently open. Do you have extractor fans fitted? Do you dry clothes indoors? Could there be a leak from somewhere?
Once you have gone through the above, and of the mould comes back I advise contacting your council and asking for a survey to be carried out on the mould? Could a health visitor possibly visit and contact your council regarding the damp and mould?
You could follow the advice from other posters and keep emailing/write to your MP, but the fact is, you will be offered permanent accommodation when you have the highest priority out of other bidders. Priority takes into consideration length of time on the list, and also banding. We had contact from a local MP occasionally, but a letter from an MP is not going to help someone push in front of others on waiting lists, or magic up a property quite simply.