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I was watching this masterpiece Memento, but one question that stuck in my mind is how does Leonard remember reading and writing? He was suffering from short term memory loss and if reading and writing was in his permanent memory then why didn't he remember his name? His name must be in permanent memory too.

Remembering skills is different from remembering facts, and memory is very complex.

Forgetting your name is different from forgetting language. For example someone suffering from amnesia might forget whether they like the taste of an apple, but they know what an apple is.

If you argue that Leonard would forget reading and writing because he learned those skills a long time ago, why wouldn't he forget all language, including spoken? Wouldn't he regress to the mental state of a baby? No. His memory of events and facts is different from his understanding of language.

For more reading, see this article on language memory and amnesiacs.

Besides, as others have pointed out, he can actually remember everything from before a certain point, so the premise of the question is flawed. That said you can't make hard rules about memory and say it's a 'plot mistake' if they are broken. Memory is complex.

This question is built on a false premise.
At no point does Leonard forget his own name. One of the first lines spoken is Teddy calling him “Lenny”, to which Leonard responds “it’s Leonard”:

(Ignore the weird doodle).

All he forgets are events that happened after his attack.

He says it himself several times in the movie: "I can't form new memories." His condition doesn't cause him to forget what happened before his brain was damaged, it only prevents him from forming new memories.