In Northumberland and Co. Durham (both on A1 and with mainline train services) there are whole townships with crumbling rows of terraced houses.
How about a bit of 'out of the box' thinking? All these terraces could be renovated into living accommodation for MPs (no more scandalous claims for second homes in London) and a new administrative centre and Parliament built.
London would still continue (you can bet your last penny that rail, road and internet links would improve) and the North would be rejuvenated and earn money for the UK. Not least, most of the MP's would realise that London was not the be-all and end-all and that there was the rest of the UK to consider.
As an historic building, I think the current Palace of Westminster should be preserved, as cheaply as possible and turned into yet another tourist attraction - but perhaps all the office space could be turned over to charities' use?
All this is off the top of my head and not properly thought through - but the basic idea seems to me to be sound. If the partners of MPs want to go to Harrod's, then make the North attractive enough to Harrod's to go there! Well, it's a thought!