Lyn would have been sending.....

This is Alex Ferguson writing, not Hilary Mantel. The former is a football manager, not a profession known for its command of language, whilst the latter has won the Man Booker prize.

Ferguson's words sound more conversational which is probably why he used them, and if he had a ghost writer he's probably being directly quoted so as to make it sound like it was written by Ferguson.

In short, don't look at autobiographies written by celebrities and sports people for examples of good English.

In your own words, how should the sentence in question read?

Thanks again for your kind help.

It's fine, I can't see anything wrong with it.

I'm sorry to get back to you but I still need your help.

What I find confusing is the use of would.

Again, many, many thanks for your kind help.

It's a colloquialism, something would have been happening while something else was going on. Two things happening at the same time in different places.

Would you please give me a couple of examples of would used with the same meaning as above?

Thank you.

Mary would have been in the kitchen when her husband was on his way home.

Lawyers would have been working behind the scenes while negotiations were going on.