How to separate sodium carbonate from sodium hydroxide?
There is a forum which discusses the similar problem of separating sodium hydroxide and sodium carbonate from drain cleaner. The solution is taken from there(slightly edited): 


  This is relatively easy because sodium carbonate is practically
  insoluble in 50% sodium hydroxide solution so a glass frit type filter
  is needed to remover the insoluble sodium carbonate. Most commercial
  sodium hydroxide contains only a little sodium carbonate from
  absorption of atmospheric $\ce{CO2}$. So for carbonate free sodium
  hydroxide solution, make up one litre at a time of say 60% solution.
  To do this, add about 300g of sodium hydroxide pellets of flakes to
  500ml of ice-cold water in a large beaker and stirr until practically
  dissolved, cover and leave to cool . Allow it to cool to about 20-25 C
  and the add another 300g of sodium hydroxide and stirr until most has
  dissolved (much less heat is generated on the second addition), dilute
  to one litre, pour into a 1 L thick-walled polythene bottle and leave
  it to stand for about 1 month in a cool place and then decant the
  clear sodium hydroxide solution.
  
  If the material is say 50% sodium carbonate do the following: 
  
  Add 500g of mixed sodium salts to 500ml of ice-cold water, stirr the
  slurry periodically for a couple of hours and then filter through a
  glass frit funnel (a vacuum filtration set-up will be needed) and wash
  the cake with a little water, say 50-100ml, to remove most of the
  remaining caustic soda; some sodium carbonate will dissolve but this
  will precipitate on the next addition. The cake will be almost pure
  sodium carbonate but you can always treat it with saturated sodium
  bicarbonate solution to ensure that it is.
  
  The clear filtrate should be placed in a calibrated vessel and another
  500g of material added and again stirred periodically for a couple of
  hours then filter and wash as before. Hopefully there should be about
  700-800ml of very strong sodium hydroxide solution. Place it in a
  polythene bottle and let it stand of a month so the sodium carbonate
  can crystallize out, decant the clear solution and dilute as required.
  
  NOTE: 
  
  
  60% caustic soda solution is rather viscous and will not
  filter without suction even on a fairly coarse frit, 50% is still
  pretty viscous and difficult to filter.
  The sodium carbonate that
  crystallises out sinks very slowly hence the long standing time.
  Sodium carbonate dissolves rather slowly in water form some reason so
  with vacuum filtation the residence time of the wash water will
  actually dissolve very little sodium carbonate, too much wash water
  will dilute the solution below 50% and make the carbonate more
  soluble.
  Hot 50% caustic soda solution is extremely aggresive
  towards skin! When handling strong sodium hydroxide solutions, keep a
  bowl full of dilute citric around in case of emergency.
  
  
  There are other method such as treating a more dilute solution with
  slaked lime, filtering (difficult because the precipitate is fine
  grained) and evaporating down.