We need a revising chamber that takes time to look at proposed legislation with as much expertise as possible and no blind loyalty to any political party. Yes, the present system is rotten to the core but replacing it with a elected Upper House would not remove party politics from that chamber.
We could have a system where every “body” which includes some expertise seek to be a “registered body”. These should include all professional institutes, trades unions and any group which includes those with similar expertise (such as a university, a federation of businesses or a chamber of trade). Registration would be needed to ensure that the group met certain requirements.
Every five years, let the names of all the groups be put into a hat (or the electronic equivalent) and the first three hundred drawn from that would have the right to appoint a UHR (Upper House Representative) to sit for the next five years.
Thus we have the required randomness and the reasonable chance of some expertise being available when required.
Finally, may I draw your attention to another blog written by my friend Lucy Brown on this matter which is well worth reading. A fix for the House of Lords.