An Informal Letter

Dear Todd and Ellie,
Sorry for not writing earlier but I’ve been very ill since I got back. The change in the weather doesn’t favor me.

I’m writing to tell you how happy I am about having visited you. I didn’t expected that the British would be such hospitable and polite people. It was a really fantastic time! I have met new people, gone sightseeing to interesting places and improved my English.

...about visiting you sounds like a future event.

Only one past-tense verb per statement: I didn't expect

such is for nouns, so is for adjectives. The British are so hospitable and polite. The British are such hospitable and polite people. If you end the phrase with a noun, you must use such.

Use present perfect in the last sentence (have met, have gone, have improved) to relate the past to now.

I will always remember about the Royal Wedding. It was so thrilling to be there. Kate and William were gorgeous. The atmosphere of London was amazing! Thank you for taking me there.

Remember needs an object (what is remembered?), so no preposition.

The wedding was thrilling (it gave a thrill); you were thrilled (you received a thrill).

-ing forms of a verb follow prepositions like for, in, at, etc

In recent days, I have been incredibly busy in preparation for my exam. This is the last year of my studies and I am trying to get the best possible results. I am doing a lot of courses and practice to improve my experience. I hope I will get a job.

I am ___ing = current action.

I must be going now. Thanks again for everything. I hope you will be able to come to Poland. The best time to come will be sunny June. My family would love to meet you.

Best wishes

Sylwia

Good work - I hope the corrections help a little. :)