First, don't give me some line about how everybody who does anything of any significance must be elected in order to be legitimate. They don't. They just don't. We don't have to elect our judges or policemen or tax collectors or jurors or head of state or top general or national football manager or legislators or other important influences in our national life in order for them to legitimately do their jobs. We could choose to elect all of these people. One might even devise various systems whereby elections for these people would work, after a fashion. But we've got on perfectly happy up to now without electing any of these people apart from some of the legislators, and it's just plain wrong - and obviously wrong - to say that everyone else was illegitimate.
Next, what's the problem you are trying to solve by having an elected second chamber? I suspect it's this: you think that having an appointed second chamber means that the Prime Minister has too much influence over who's there. How is that solved by having a load of people elected there who are standing on behalf of the Prime Minister's party? Now, I'm guessing some of you will say: "Ah, but the second chamber will be elected by PR, so its make-up will be different from that of the Commons." But you're chasing the problem now - you wanted to reduce the Prime Minister's power to select who is in the second chamber by increasing that power (making them his elected supporters), and so try to compensate by fixing the election through PR! But why not just have that the membership of the second chamber isn't up to the Prime Minister at all? Why must it be the case that the Prime Minister has any say over it? You're trapped in the vision of the Prime Minister as some all-powerful dictator, entitled to do anything just because his party won a General Election.
Obviously you can't prevent the Prime Minister from determining who is there if you have elections, so if you are really serious about limiting his control over the membership of the second chamber, you blatantly can't support an elected second chamber. There are lots of things you could do instead - there could be appointments by someone else (e.g. the Queen); people could be selected randomly (like a jury); people could be sent there as delegates of other organisations, such as businesses or religious groups; people could inherit the right to sit there; people could buy positions there; people could win positions there by buying tickets in a national lottery (or buying chocolate bars, Charlie Bucket-style). Obviously each of these concepts (and combinations of them) have their own strengths and weaknesses. But they all share the feature - which neither appointment by the Prime Minister nor election has - that Prime Ministerial patronage is reduced.
Also, why have a second chamber at all? In other jurisdictions with two elected chambers there is typically a rationale. For example, in some federal systems the lower chamber consists of deputies selected from local constituencies whilst the upper chamber is selected at the level of the member states of the federation. What would be the rationale for having two elected chambers in the UK?
I presume that the answer is going to come back: it is useful to have a revising chamber. But why a chamber? Why not have special committees that consider legislation in detail and at length and propose revisions? Why do we need extra elected people to do that? If you insist that the committee must consist of elected people (for which I see no good reason), then why not have the committee made up of members of the Commons?
(Indeed, given that most legislation comes from Europe anyway, there is a pretty strong case to be made that even the Commons is mainly just a revising chamber as things stand. Do we really need two revising chambers?)
Someone else might say that the second chamber is there to provide some sort of inertia, e.g. by elections occurring less regularly than in the Commons. So, are you saying that you want something that contests the will of the main elected chamber? A kind of check or balance? Something that thwarts the will of the mob, because sometimes under democratic pressure elected officials act in haste and in unpleasant and ignorant ways? Golly! You're sounding suspiciously like what I might once have called a "Conservative" there. Be careful - the modern Conservatives might hear you!
But let's run with that. On this thought we are not Democrats - we do not believe that the Will of the People should always prevail. Sometimes what is just and proper and honourable and in everyone's long-term interests will be not to do what the People want. That being so, I am mystified as to why you would want elected people to protect us from being too sensitive to the pressure placed on elected people. If you want mechanisms to protect policy-making from over-sensitivity to the Mob, then surely you want people that do not owe their position to votes? Can't you see that, as in the case of the PR proposal above to limit Prime Ministerial patronage, the use of single terms and long periods of office are means of chasing the problem - a problem created by electing these people in the first place?
Is it really credible that, even with long single terms, an elected chamber would offer any real protection at all, here? Even if those in the second chamber were only permitted to stand once, the only people elected would be committed Party men and women. (And doubly so under PR.) They would make their decisions recognising the influence those decisions would have upon up-coming elections in the lower chamber.
If you are going to have a upper chamber with any point, it needs to have its membership determined in some fundamentally different way from the lower chamber. I believe that a constitution that limits and steers the democratic will in a number of ways is likely to be more liberal, more stable, and more just than a system that surrenders entirely to the principle that what people want is what should happen. One of the most important such limits on the democratic will is having a second chamber of legislators that is not elected. Even under the current flawed system, it is well-recognised - a truth understood by top policy-makers that dare not speak its name - that the House of Lords has in recent years been a key delaying point for much of the most misconceived, knee-jerk and illiberal legislation brought forward.
That doesn't mean I like the way the House of Lords' membership is determined now - far from it. But not liking the current system does not imply that just anything must be better. And an elected second chamber isn't.