Learn Icelandic: Kæri Jón, þú ert innblásturinn minn.

Even though there is no definite article or pronoun before it, adjectives used for direct address (when you are talking or writing directly to someone) take the weak form. This is why we use the weak masculine form kæri when addressing Jón.

If I were writing to a woman named María, how would the address change?

You would use the weak feminine form of the adjective, which is kæra. The greeting would simply become Kæra María.

Why does the sentence use innblásturinn minn with the definite article, rather than just innblástur minn?

In Icelandic, when you use a possessive pronoun like minn (my) after a noun, the noun almost always takes the attached definite article. So innblástur (inspiration) becomes innblásturinn (the inspiration) before adding minn.

What gender is the noun innblástur?

It is a masculine noun. You can spot this easily in the sentence because it takes the masculine attached article -inn in innblásturinn, and is followed by the masculine pronoun minn.