I will try my best to finish it and won't betray your trust.

我会竭尽全力去完成，不会辜负您的信任。

However difficult it might be, I will try my best to work it out.

不管有多困难，我都会尽力解决。

If you have other requests, please feel free to write to me. I will try my best to meet your needs.

如果您还有其他要求，请随时写信告诉我，我会尽量满足您的需求。

B: Thank you. I'll try my best.

谢谢，我会尽力的。

But I will try my best in London.

但我会在伦敦尽最大的努力。