Tikkun Olam is a great idea. It's just not in Alenu.

Re. the correct spelling of le-taken olam in the Alenu prayer (mistakenly translated in this context as "to repair the world)," Mitchell First has written convincingly on this topic. The earliest textual evidence certainly points to לתכן ("to establish") instead of the prevalent לתקן ("to repair").

I would add, however, that the strongest proof for לתכן is contextual rather than from textual variants -- that is, from the Musaf prayer itself and from related biblical verses from which Alenu's author seems to draw.

The expression לתכן עולם appears to allude to the second Ketuvim verse (Ps. 93:1) in the Malkhuyot section of Musaf: 

ה' מָלָךְ גֵּאוּת לָבֵשׁ לָבֵשׁ יְהֹוָה עֹז הִתְאַזָּר אַף־תִּכּוֹן תֵּבֵל בַּל־תִּמּוֹט

More precisely, לתכן עולם במלכות ש-די is a compound phrase constructed from the last words of that verse -- לתכן עולם follows תִּכּוֹן תֵּבֵל -- combined with the biblical usage תכון מלכות (e.g., I Sam. 20:31; I Kings 2:12), the establishment of a permanent dynasty or kingdom. 

But no matter how you spell the word in Alenu, the current usage of Tikkun Olam represents a noble effort that, in manageable portions, is much more practical to implement than bringing God's kingdom down to earth. Quite possibly, the two programs are effectively the same.