I'll say you can.
Unelected head of state
Unelectected 2nd chamber
Unelected cabinet
Unelected Prime Minister often. Dont forget when Theresa May stood down, Johnson was elected Prime Minster by less than 200 000 Tory party members, including 16 & 17 year olds. That's 200 000 in a country of 66 million
Unelected special advisors who seem to have a large say in how this country is run.
Then when we do have elections party funding means that billionairs have a big say in policy for no other reason than that they happen to be stinking rich.
A head of state who has taken no part in the day to day running of the UK in our lifetimes.
Unelected 2nd chamber isn't ideal, but rarely do they block legislation. Generally they send controversial bills back to commons with recommendations. It's also quite likely that we see an elected 2nd chamber before too many more years pass.
The cabinet are all elected members of parliament. This is how governments are formed in a democracy- either from the majority party or a coalition of parties, where no majority exists. Frankly, the inclusion of this point is ridiculous.
Prime Minister is also elected to parliament. No one votes for the Prime Minister - they vote for regional party representatives who may form a government either by majority or by means of coalition. Consider that is entirely possible for an incumbent party leader to fail in their effort to be elected. Under such circumstances, that party must then choose a new leader. Your list is taking on the appearance of a handful of straws.
Unelected special advisors who advise. You realise that all leaders must seek advice from somewhere? Politicians are human beings who can not possibly keep themselves abreast of every single issue that may require government time. Would you rather that governments sought advice from Mabel at number 32?
The funding of political parties is certainly an issue, but we have a transparent system that requires all party donations over £7500 to be declared and donations over £1500 where finances aren't managed directly by party HQs.
All told I think you've helped make my point, so let's look at the other side of this coin...
Here's a list of primary EU institutions.
European Parliament
European Council
Council of the European Union
European Commission
Three out of these four are entirely unelected by the people of any member states.
The exception is that wonderful European Parliament, where MEPs can choose to vote in favour or vote against legislation, after a brief period of discussion. No MEP can table ammendments to bills. No MEP can introduce bills. Decades of European Parliament business has demonstrated that bills rarely fail in this parliament and those which do fail, tend to be resubmitted and subsequently pass. Perhaps a reasonable mind might ask why this is - that same mind is unlikely to conclude tha the unelected bureaucrats that make up the EU Commission are so prescient that they know what is best for almost thirty individual countries on every single occasion.
Maybe we should begin to list permanent members of the EU commission and document how many were political failures in the member state from which they came? Time constraints mean I won't do that right now, but I will remind you that the current commission president barely scraped through her election, despite being the single named candidate on that ballot paper - just think about that for an example of democracy in action!