It should have installed.
What kind of SD card are you using? Cause anything 1GB or 2GB higher is not recommended.
Is the Super Smash Bros. Brawl game for your region. Rentals should work, but an actual copy from a major store would be the best (my disc was bought from a gamestop used and cheap, returned it after I did the installation and got my cash back)
Make sure all the custom stages are deleted.
Back out to the menu (or start screen, I accidentally did that), insert the SD. Go back into stage builder, the game should read your SD card and the installation should proceed.
is the wave of the future!