This is regarding the soluability of a chemical and how to change its soluability.

With the exception of many of the non-metals, I can't think of any ELEMENTS that are soluble in either water or nonaqueous solvents. (OK, many metals dissolve in mercury) (OK, OK, they also may well mix if melted)
Complexation is one way to solubalize a COMPOUND. For instance, AgCl dissolves in aqueous ammonia because of solvation of Ag+ to Ag(NH3)6 +. Many other complexing agents are known.

On further thought (I got hung up on the "Fat soluble element" part.) this is a classic problem in Organic Chemistry. Say you have a mixture of an amine, an organic acid and a hydrocarbon and want to separate them. All three are soluble in ether, none are soluble in water. To effect a separation, you would extract the ether solution with dilute HCl. This changes the amine into it's water soluble salt (RNH2 + H+ --> RNH3+) and it is extracted into the water layer. The remaining ether solution is then extracted with dilute NaOH. This changes the acid into it's water soluble salt (RCOOH + OH- --> RCOO-) and it is extracted into the water layer.
The solution with the amine salt is then made basic, the solution with the acid salt is made acidic and both are re-extracted with portions of ether. The result is three ethereal solutions with the separated components in their respective places. The ether is evaporated and all three compounds are obtained as pure compounds.