Laplace Transform without given continuity variables
The circuit as shown is impossible because the 2A current source (Item #13) is open circuited, thus producing an infinite voltage across the gap.  There must be another switch which shorts this current source which then must be opened while the other (shown) switch changes position.  If the #13 current source is intended to be connected across the inductance, then IL=2A at t=0- and t=0+.  But I wouldn't waste any time trying to solve this bogus circuit until clarification is supplied.

A current source can NEVER be left floating, it must be shorted when not in use.  The dual to a voltage which must be left open when not in use.