To avoid very inconvenient emulsions, be sure SWIY's pH is 12+... an excess of NaOH will accomplish this, and then it is safe to shake vigorously.
Warming the extraction seems to make it a bit more efficient. Like Soulfood says, pull till it's spent... then look into pulling with a different solvent (Xylene, for example), and salting out the last of the last!
Bon voyage!
L
"The first principle is that you must not fool yourself, and you are the easiest person to fool." Richard P. Feynman