2 or 3 or 4 times, repeatedly, with soap such as Ivory Soap or Woolite. Using a detergent like Tide may dissolve the indigo somewhat due to its alkalinity. The longer answer involves chemistry and history. The advice to set the dye with vinegar or salt is commonly heard and seems to make sense from the standpoint that indigo is mildly soluble in a reducing solution, insoluble in an oxidized state. Saltwater also has nothing to do with the chemistry of indigo dye.
Peroxide is essentially super-oxygenated water.
Think of the excess indigo as blue fuzzballs or chalk dust. Something that does not blend with the fiber itself. In real life, and with a cotton hakama, it is far better to just wash the excess out before it falls out in chunks all over your mat and furniture. Modern methods of indigo dyeing involve a chemical reducing solution which simply cannot get the dark dark blues that the older methods did. If youre trying to clean a mat be sure to test it on a small, inconspicuous and low-stress side spot of the mat first to make sure it doesn't take everything else off too.