The phrase "of which" is usually used in non-defining or non-essential relative clauses, which are offset by commas, and which are not essential to identifying the particular antecedent.
If the context of the sentence makes it clear what particular plane is being referred to, the sentence is correct if you add the commas:
The plane, the right engine of which broke down, has been repaired in Cairo Airport.
However, if knowing that the right engine broke down is in fact essential to identifying the plane in question, the relative clause is essential, and must use a relative pronoun (which will require some rewording):
The plane whose right engine broke down has been repaired in Cairo Airport.
The plane which had a broken-down right engine has been repaired in Cairo Airport.