*Your issue: Complaint to about/concerning the flight attendant 

Dear Sir/Madam, 

I am writing to complaint  complain about my dissatisfaction with the conditions on my return flight from London to the Canary Islands. The Incident happened  took place one week ago, on the 21st of January.

took place = more formal than 'happen'. 

Complaining about my dissatisfaction = sounds quite strange. It is like you are complaining about your own dissatisfaction, like it is your fault. You need to choose either:

I am writing to complain about the conditions

or

I am writing to express my dissatisfaction with....

I had been informed that there was gluten-free food on the menu. I am celiac so that I ordered to  asked the flight attendant to give me a gluten free sandwich. After that, I had started to eat the sandwich when I discovered that it wasn't gluten-free.    Only after starting to eat the sandwich, did I discover that it wasn't gluten free. 

Here your tenses have an issue. You could write 'After I had started to eat the sandwich, I discovered that it wasn't gluten free'. but because you used the words 'after' and 'when' together, the tenses sounded odd. 

Then, I informed requested to the cabin crew about it and they told me that I was my fault because I didn't check the label beforehand. I was extremely upset by this, especially because I felt sick during the rest of the flight. 

Request + object e.g. 'I request better conditions'. You can't request about something. 

I feel that airline should take responsibility for this incident. I am therefore requesting compensation for the flight, which cost 200€. I also trust that you will investigate the situation to ensure that this does not happen to other passengers in the future. 

I look forward to your reply. 

Your faithfully, 

César Snow

Good job.

Pay attention to indirect and direct verbs

e.g. Request something from someone 

Order to = order

I order you to do something (although this is too strong, (like a military term) so i replaced it with ask)