The whole point of HoL is to be a proper revising chamber. I recall my Constitutional law tutor always said the problem in our parliamentary democracy was reform of our HOC and not the HOL. Get that right and HOL will follow. Problem is HOC does not want to give up or be slowed down or have it's legitimacy undermined by a legitimate upper chamber.
An upper chamber is could be elected by expertise and serve set limited terms.
Which comes back to whether we should have proper separation of powers in our constitution. With the executive branch properly sitting outside of a legislative lower and upper chamber. With a upper chamber properly representing the four nations. And that would probably mean we could then deal with what is point of the monarchy? And a written constitution. And an elected President whose role is to be guardian of the Constitution with the PM and cabinet serving that office to execute executive powers.
But god forbid a federalist structure. Better to have a hybrid Parliamentary democracy with devolved powers like we have at present.
House of Lords is a joke. For the few people who actually do their job well there are so many members who do almost nothing and collect their allowances. Both Conservatives and Labour spend their time stuffing the lords with more members to try and get a majority that it's just become bloated and pointless. Hell the conservatives just stuck their ex-MPs in there when they lost their seat (Zac Goldmsith). For me the House of Lords represents a past that for some reason British people want to cling onto as it's part of our history, but actually what it represents should be completely abolished in a modern democracy. While the majority are appointed by parties, some still are genuine lords with ***les and hereditary positions. Why should people in this day still be given positions based on birth? I can at least understand the argument for the Royal family as a tourist attraction but the rest of these nobles that still exist are merely wealthy aristocrats who inherit advantages that others would not have access to. Why should these people have a say in how our country is run just because of their family? Our democracy based on tradition has shown to be flawed and with the blatant corruption and cronyism happening at the moment we no longer have the moral high ground over other countries (personally I don't we ever did, but most people in this country would think we do). I reckon they'll find that Cameron should have acted better, but didn't break any clear rules. However for me, all that says is the rules are not fit for purpose. Most of what the government has done has been within the rules, so clearly the rules aren't good enough to effectively police politicians and their behaviour. Overall we have an outdated democracy based on traditions that are no longer fit for purpose or even relevant in some cases.
For me we first of all need a clear constitution that has clear procedures and processes for how democracy in this country should work. Recently we've had potential issues and conventions from the 1800s being used. That's unacceptable and there should be no ambiguity. How it should be written? Ideally with as little influence from politicians as possible. It should be a neutral document that allows for a fair playing field. Need a group of experts from a wide range of professions and backgrounds to represent the whole of the United Kingdom.
Second FPTP needs to go. As long as we have that we will only ever rotate between the two main parties. There was a brief hope with the Lib Dems in power that the traditional system would break up, but that failed due to Lib Dems being useless and the fact that politics has become even more polarised. There is now no decent political debate. It's become vote for us because the other side will do this. There are also aspects of Americans tribalism where people just vote for their party regardless as that is their team. The people who don't vote like that vote for the party that is most likely to stop the party they don't want. Our politics is so negative and instead of getting the party that will do the best we get the party that will do the least worst. I don't think proportional representation would work with constituencies, so I would go with a transferable vote system. That way people can vote for the party they still think is the best and also still vote to keep the party they least like out. Whether it would be a single transferable vote or multiple transfers is debatable as each has merits and negatives.
HoL needs to be abolished and a new upper house without the taint created. This for me though is more widely open for debate. Should it be experts in fields with no political agenda (hard to find)? Should it be political representatives with some kind of proportional representation (how do you stop it become HoC v2?)? Should there be a mixture of different representatives serving different terms? (How can this be kept fair? Example would be the U.S supreme court?) Honestly I'm not sure right now which would be best.
Next papers should be banned from supporting political parties or rules need to be tightened so they make it clear where their bias lies and this should apply to all times, not just elections. I don't think Corbyn would have won anyway, but there is no doubt that he was targeted by a media smear campaign. Our politics is too heavily influenced by biased media and while some is on the left, the majority is clearly on the right.
Finally we need to address the issue of boundary changes. Too often it's become political and should be completely independent and unbiased. Again, the how is tricky, but we need to make sure that we never get into a situation like in America with gerrymandering.
Edit: Also more clearer and stronger rules to stop corruption & cronyism. The government shouldn't be able to ignore this and claim that the people will decide in 4 years. Voting should reflect the legitimate work and accomplishments of the government, not whether the country thinks they are corrupt or not.