I'm a psychology graduate. I need your advice!

So I have graduated from a top 5 university in the UK. I'm now taking a gap year off before going back to a graduate school. Upon my graduation, I have kind of changed my mind that I want to do investment banking for now. However, I haven't got a single internship in banking during all three years of my study. All I got were couple of internships in research and some roles in charities (your typical "ideal profile" except that finance experience).
My educational background: A levels in maths, psychology and history (range from A to A*)
I got a first class degree.
Do not have any finance related knowledge AT ALL.
Here are my questions:
1) Do I still have any chance to break into banking?
2) Should I do a two year master so that I have a summer in between to do an internship?
3) I'm thinking about doing a master in management. Can I get into banking with this degree? (I don't think I can do a msc finance at this stage. I'm not bad at maths but I just don't think I have a good foundation to cope well with the course given my educational background. I know that working in banking doesn't need to know differential equations no?)
4) What should I do this year to prepare myself? Can I even get a banking internship even I have no finance knowledge? Will an unpaid internship in other business related fields like human resources help?

I would greatly appreciate all your advices! I'm now experiencing a bit of quarter-life crisis here.