I found some information on the difference betwee washing soda and soda ash on a website:

http://www.welltellme.com/discuss/index.php?action=printpage;topic=20783.0

The discussion was primarily directed toward people using soda ash in dyeing cloth, but I suppose the information on the difference between the two – washing soda and soda ash – would apply to laundry soap making, too. 

            “Soda ash is the active ingredient in washing soda. The chemical name for it is sodium carbonate, chemical formula Na2CO3. It is more basic, that is, less acidic, than sodium bicarbonate (baking soda), whose chemical formula is NaHCO3. The purpose of sodium carbonate is simply to increase pH.

Hydration

Some forms of soda ash (e.g. that labeled as 'washing soda') contain more water molecules than others, which makes them weigh more and be larger for a given number of sodium carbonate molecules – this means that you need to measure out a larger quantity of the hydrated form in order to get the same results. If you buy sodium carbonate without the extra water molecules, then store it for several years in humid conditions, it will absorb some water and appear to lose strength, when in fact it has merely 'bulked up' and needs to be used in larger volumes.

(My own emphasis added:

The type of sodium carbonate used in washing soda is a decahydrate, so you need to use a lot more washing soda than you would anhydrous soda ash, perhaps three times as much - assuming that you are able to find pure unadulterated washing soda that is suitable for use in dyeing. (In theory, we should use 2.7 times as much washing soda as a substitute for soda ash, if measuring by weight, or 4.6 times as much if measuring by volume.)

Where can you buy soda ash?
Although you can buy washing soda in the grocery store, this is usually advised against, because some US brands in the past were known to contain optical brighteners, salt, and/or fragrances. However, Arm & Hammer brand “Super Washing Soda” does not now contain any additives, though you must use a larger quantrity due to its greater hydration level (see above).

A better source is a swimming pool supplies store, or a hardware store that carries some pool supplies, which will carry pure sodium carbonate (a popular brand is pH Up), sold for the purpose of increasing the pH of pool water.

This is typically cheaper than mail-ordering from a dye supplier, and just as good. Beware of suppliers that try to sell you sodium bicarbonate instead of sodium carbonate; bicarbonate is much less basic (it has a lower pH), and will not work for most dyeing recipes.”

From this information it seems that soda ash (found in a pool supply store) has the same chemical properties as a super concentrated washing soda. It sounds like you could use soda ash in place of washing soda in a laundry detergent recipe, but use much less (at least 1/5 the amount) of washing soda.

I'm no expert. I have no idea if this information is accurate, or safe, but I found the same basic discussion on washing soda versus soda ash on more than one website. The site I referenced above seemed to have the most usable information.