First, let me assume that he means Methyl Isobutyl Ketone for MIBK and that the solid in question is an organic polymer, maybe even a cellulose type of material.  Also, it sounds like he has tried re-slurry washing and is using the methanol as an anti-solvent to displace the MIBK.  Re-slurry washing is generally a very good method so I have to conclude that the MIBK is binding to the polymer with a very high activation energy.  Two suggestions come to mind:
  1. Try a different anti-solvent, such as acetone.  I realize this is much more volatile and he may not want the vapors to deal with, but acetone can be one of the precursors to making MIBK and should bind much better.  The acetone could then be displaced by methanol for the drying step.
  2. Try re-slurry washing using ultrasonics.  This may require a more dilute slurry to get the ultrasound to penetrate the slurry, but it may overcome the high activation energy.

He didn't mention temperature or whether water would be damaging to the solid, but MIBK forms an azeotrope at about 190 F with about 24% water mixture, even though the solubility in water at lower temperatures is minimal.  The methanol may be too polar for MIKB and a non-polar solvent would work better.  I don't know if methanol was chosen over other anti-solvents, but non-polars have greater solubility (i.e. oils).
 
In terms of equipment choices over re-slurry washing, this will depend on what form and particle size he ultimately wants or can accept.  It sounds like the current solid is very fine and maybe even colloidal.  In which case he may want to look at high temperature pervaporation to drive the MIBK over a membrane.  If the solid is large in size the MIKB could be held in the lattice of the solid (i.e. from a co-crystallization or an inclusion).  In this case grinding or wet milling of the solids could release the MIKB into the methanol.