The monarchy is a red herring in this case, I think. You seem to assume that in the UK the monarch has more power than in reality. In truth, she has a lot less power than the US President. In the country's history people (and Kings) lost their heads to limit that power. No danger to democracy there - whereas in the US... but that is discussed in other threads already.

As for the business with the Lords, well it is changing. What happened in 1999 is step 1 of several, I think. But again the problem is not there, in this case, it is in allowing a Minister (a Department Secretary in the US) to amend or repeal an existing law without being checked by the House of Parliament/Representatives or House of Lords/Senate.

A working Democracy of whatever shape has to have checks and balances. It's when these checks go that danger exists.