Look at picture at: http://lwp.smugmug.com/photos/574024162_oYXG8-O.jpg

This is the view looking up at the switchbacks (numbered in the photo), with the "chute" on the north side (right side in photo). You can clearly see tracks in the chute. Both paths lead to Trail Crest. The top is another 2 hours (or so) from Trail Crest along the back side of the Sierras. The chute does require crampons, ice Axe, and self arrest skills. The chute is steep. Failing to self arrest would ruin your day.
See fairly recent trip report here: http://www.whitneyzone.com/wz/ubbthreads.php/topics/30859/Mt_Whitney_Trail_Conditions_Po#Post30859, then follow the link within this post.

Much more information is available on the web site. Click a few buttons.For instance, check out the following link. Several very good pictures of the chute found in NYker's trip reports. (found within the following link).

http://www.whitneyzone.com/wz/ubbthreads.php/topics/31079/Mt_Whitney_late_spring#Post31079

In two weeks you may be able to just hike via the switchbacks, and perhpaps microspikes would be sufficient (or even not needed). Keep checking. There are trip reports posted every week (although not always under "trip reports". Sometimes a new topic is added that reports on conditions.