No offense, but you are obviously an 'Am HaAretz Gamur MiDeoraitha.
Mordechai was not right-handed. He was left-handed. This is evident two Pesukim later, where he is referred to as אֹמֵן אֶת-הֲדַסָּה, הִיא אֶסְתֵּר. He was Hadassah's personal artisan, which implies not that he was her caretaker (I scoff at your naivete, sir, for thinking so - no offense), but that he was her personal makeup artist.
As the Midrash teaches us (Bab. Meg. 13a), Esther had green skin, and, as the name Esther implies, Mordechai hid this with his artistic skillz. In fact, he had such mad skillz that he was able, as in the aforementioned Pasuk, to totally conceal her identity of Hadassah, allowing her to assume a new identity as "Esther". As is well documented1, most artists are left-handed (including artisans of this type2).
The Pasuk in the question is, of course, referencing Mordechai's status as G-d's right-hand-man, the person who would bring about the salvation of the Jewish people (remember, it was a hidden miracle, which is why HaShem's name doesn't appear in the text). Further evidence of this is the fact that the right-hand is not supposed to know what the left-hand is doing, which is why Esther grew weary of Mordechai's repeated attempts to contact her inside the royal palace and stopped speaking to him directly.
Again, no offense.
1 It's not.
2 This is almost certainly false.