He does. His comments to Tikkunei Zohar (Vilna printing, pg 4):
דעסקין באורייתא לשמה - ר"ל בשביל השכינה לייחד אותה עם התורה והענין של לשמה שלומד ע"מ לקיים מצוותיה של תורה והמצוות הן בשכינה א"כ הוא לשמה
"Who study Torah lishmah" means for the shekhinah, to unify It with the Torah. And the idea of lishmah is that he learns in order to fulfill the mitzvos of the Torah, as the miztvos are the Shekhinah and therefore it is lishmah
Don't ask me to explain the above paragraph because I have no idea what it means, but it's there.
In addition, there is the more famous definition of Torah Lishmah given by the student of the Vilna Gaon, R. Chaim Volozhon, in his book Nefesh Hachayim:
אבל האמת כי ענין "לשמה" פירוש, לשם התורה... כגון לידע ולהבין, ולהוסיף לקח ופלפול
Really, the idea of "lishmah" means, for the sake of the Torah (itself)... such as to know it and to understand it, to add teachings and depths
This wasn't necessarily the Vilna Gaon's position, but considering R. Chaim Volozhoner's status as one of his greatest students a study of his position should probably take R. Chaim's view into consideration.