What would a written British constitution look like? Wouldn't it just be sections copied from various existing laws anyway? How would writing it out in one document improve things?

The House of Lords is still a thing because there's no will to abolish it. Why would we? Our voting system means that we tend to end up with a single party having a majority in the House of Commons and thus its important to have an upper house to keep them in check. Having the House of Lords be directly elected wouldn't be popular as that would mean it would just be a duplicate of the lower house. I admit that having members who are appointed for life is a double-edged sword - on one hand they can go against the government without fearing for their position but on the other hand they can be unaccountable - but it's a system that seems to work for us. Also, what would we replace it with?

I don't really have an answer for your last points other than to say that the sound rather philosophical to me. The political system in the UK absolutely needs reforming (if it were up to me I would change our voting system and phase out the last few hereditary peers and Lords Spiritual and then perhaps look at some sort of federal system) but I don't feel like the government is in any way illegitimate.