How does one assess the amount of cake or cookies that would be the equivalent of a bread meal and require washing?
It is controversial how much cake or cookies this really is, some say that it is about 250 grams, (about 8 ounces), but others say that it is more than that. Therefore, the best thing is just not do bring yourself to such a situation and eat less than that in one sitting. If you will eat less than that amount, and then wait about ten minutes, and then continue, you would not have to wash and bentch. A practical way for you to calculate things, is the following. 250 grams is 4 eggs worth, which would be 8 kzaysim (olives worth). There is a wonderful sefer called “The Halachos of Kzayis”, by Rabbi Pinchos Bodner, which has lots of pictures of different foods, (it might be hard to look at this sefer on a taanis) and how much of each on is a kzayis. You could look in the sefer, see what is a kzayis, and then 8 times that is 4 eggs worth.
O:CH 168-6,7, Poskim.