were the blocks heat set after they dried???? even with a laser printer the ink needs to be heat set- if it is not it will rinse out-
the reason for not using a laser printer is due to them getting hot- jamming- other issues like that-not because the ink is not as permenent-
it does make a difference if good (type/brand) cartridges are used as opposed to refills- generic cartridges.
as for the (bleeding?) that is probably from handling them before they were heat set.
many of those transfer sheets are made for wall hangings- not for used quilts that will be laundered.
if you have some pictures that are not too (marred) i would heat set the pictures=then try to remove the excess dye by rinsing. unfortunately it sounds as if you may have to start over. i have had much better luck printing on muslin (pre=washed- no fabric softener) pressed to freezer paper & ran through my printer- i then set them aside for 24 hours for them to 'cure' dry then heat set- the quilts have faded a little over the years- but are still quite nice 5-6 years later.
sorry for your plight- good luck.