Learn English – Any or every age

As a verb, result takes from, not of. (in fact there are very few verbs that take of for one of their arguments - I can't think of any)*.

As a noun, result usually takes of, rather than from.

Edit: Fumblefingers pointed out that in saying I couldn't think of any examples, I used one! But on reflection, I don't think it is a counterexample: think of is a phrasal verb, with a different meaning from think.

"Retiring" as a verbal noun (a 'gerund') is only used in the sense of "the process of stopping work", whereas "retirement" is used more often of the resulting state.

So when you talking about the act

I'm thinking of retiring next year.

is normal.

I'm thinking of retirement next year.

is possible, but to me means that you are thinking of the state of being retired, not of the process of retiring.

But

I'm going to enjoy my retirement.

You can't use "retiring" there.

Clearly this isn't the whole story, because it would imply that, in your example, we would say "the legal age of retiring". But, while this is possible, it is not natural. I think set phrases all use "retirement": "age of retirement"; "retirement party"; "retirement present".

As others have pointed out "retiring" is also a participle and an adjective.