I actually think that there are benefits to a monarchy, albeit not an absolute monarchy. We've seen how politicians often focus on the short-term so they can get re-elected, sacrificing any chance for long-term planning. A monarchy would be immune from this, and more concerned with the long-term welfare of the nation. Of course, you'd need an elected legislature to give the people a proper voice.
I don't think this necessarily requires a monarchy. You get much better long-term planning in coalition democracy, because all of the parties are forced to negotiate and compromise. Therefore, opposition for the sake of opposition is politically impossible, not a benefit (this seems to me to be the primary mechanism through which you get short-termism). The other advantage of coalition government is that it's far more democratic.
Another proposal: Upper Houses should be appointed, not elected (this is basically true in the US already, due to the auto-gerrymander of the Senate). Some group - like, but not necessarily, the government - appoints a new member of the upper house whenever one dies or quits. These people should be eminent scientists, academics and so on - maybe set up a quota of ignorant businesspeople tories are allowed to appoint. Once they're on, they can't be kicked off except by a judge (if they're corrupt). These guys have two roles. Firstly, they can delay legislation. If the government tries to pass the We Win Act of 2014, the upper house can delay it for X long. In extreme cases, the upper house can also force the government to put the legislation to a referendum, to give the people the ultimate choice.
The other thing the Upper House should do is hold loads of committees. Parliamentary committees are actually very useful - you can demand any document, with power of law, force anyone to testify. There would be no danger of the partisan manipulation of the committee system ala Huckleberry Graham et al because none of these people are partisans. Therefore, the government can only act secretly where an external, independent body thinks it ought to be able to. If the government keeps something from the upper house and it leaks, there is an instant election and the responsible ministers are thrown in a dungeon.