I've not applied for Letters of Administration but I have gone through the probate system and it's basically much the same. I found everything very simple.
The Gov.uk webpage mainly refers to 'probate', rather than to 'letters of administration', but you can use the same online service, or paper forms, in either case. (My experience of the system was before the days of online applications. I found the paper documents quite straightforward anyway):
https://www.gov.uk/applying-for-probate/apply-for-probate
(Other AB members have used the online service, with generally very positive feedback on the posts that I've read from them).
I found the people at my local Probate Registry incredibly helpful when I went through the process. These days there's a national helpline number which, if it's anything like HMRC's national helpline, might be difficult to get through to. You might find it easier to call your local Probate Registry directly.
The national helpline number for new paper applications is 0300 123 1072. The number for ongoing applications, or for help with digital ones, is 0300 3030 648.
The postal address of your Probate Registry can be found using tis facility:
https://www.gov.uk/find-court-tribunal
(You'll only find the national helpline numbers quoted there but Google finds local office numbers).