What are my chances of getting into LSE Econ without further maths predicted grades

My sixth form doesn't offer Further Maths, therefore I am planning to do it externally. However, I won't have predicted grades for it or anyone for reference.

Should be fine. You'll need to list that you're taking the qualification (with a grade of "pending") and your academic referee should note you're taking it externally and they aren't able to provide predicted grades. That should be sufficient.

okay thanks. Is there anything else important that I need to know that will increase my chances when applying for uni

There's no magic bullet you can get that will guarantee you admission over others. You will need to do wider reading but there's no specific text which will "make you stand out" over others, as they're not looking for you to have read anything specific but rather on the quality of your analysis and reflections on what you have read; there are various other opportunities you can partake in that might be of relevance but again, what you did is less important than how you analyse and reflect on it in your personal statement, particularly as they recognise not all applicants have equal access to e.g. work experience, competitions, etc.
The best thing you can do is start thinking and acting like a university student - read beyond the curriculum, think about what you've read, write carefully and critically about it, take up opportunities as they present themselves, and don't think of things in terms of box ticking assessment objectives and more about your holistic intellectual development in your proposed subject area.
Obviously also, make sure you prepare thoroughly for the exams you do have to do as well as you can in them.

Thank you so much, I'm really scared about the personal statement part, as I've never really been good at writing unless it is analytical, therefore I think it might be my weakest point. However, I am quite confident that I will get good grades as I am good at studies and well dedicated.