Cambridge English: First

As the instructions state you should write to Mrs Sykes, you need to say 'Dear Mrs Sykes' or 'Dear Madam'.

An excellent start to the letter. However, try to avoid using contractions (I'm) here and elsewhere in this answer. The examiner will be looking to see you appreciate the difference between formal and informal writing and the use of contractions is usually found in more informal texts.

What type of club? Give the reader of your letter more information, e.g. 'a similar club' or 'a film club'

This sounds a bit unnatural. What about saying something like: '...which demonstrates my organisational abilities.'

Cut 'Also' this is not necessary

Use a semi-colon here instead of a comma. 'I have been studying English for a few years; I got a merit....'

Use another way to begin the sentence so it's not the same as the previous one, such as 'In addition, I am...' or 'Furthermore, I am....'

I suggest you join this sentence with the previous one to make it a single paragraph.

This would be better as 'a suitable person'

You could also say 'be pleased/delighted/interested to hear from you.' (Passive constructions sound slightly better in this context)'

This should be 'faithfully'