Connection issue for dual LED utility light
Power strip vs extension cord make no difference.  They are the same thing.  Generally you are not allowed to use flexible cords as a substitute for the fixed wiring of a structure. 

However, NEC 400.7 specifically allows the use of flexible cords for luminaires.  The minimum permissible wire gauge in such a lamp cord would be 16 AWG.

I don't know if this exception extends to extension cords.  I suspect it does not. 

There is no problem using a longer cord.  There's no maximum cord length in the NEC.  However you must install it properly with good strain reliefs.  If you don't know where to find a longer cord, go to Target in the extension cord section and do not leave the section until inspiration strikes!  

Check the writing on the cordage to determine its size and type, and make sure this is a type legal for use as cordage.  Most likely it will be. 

I would not recommend soldering it to a board.  If there is a metal-enclosed place on the lamp which is suitable, I would leave several inches of the old cord, and use wire-nuts to splice the old cable to the new.  Mind polarization: the edge of the cord will have ribs or a stripe, which goes to the wider blade on the cord, aka neutral.  Polarize the new cord the same way.  This splice must happen inside a metal-enclosed part of the lamp, and it cannot interfere with the strain relief protecting the cord from damage.  Without a ground wire, it's especially important you don't cut open a wire or short it to chassis.