Leaf Miner out of control

Stentman, what I'm doing is to check the plants daily and rub off the little clusters of white, thin eggs left by the leaf miners, usually on the lower part of the leaf. It seems to be stopping the invasion... You might damage the leaf a bit since the eggs seem to stick strongly to the leaf, at least if you don't find them immediately, but it's anyway less damage than the miners will do if they get out. At the beginning I kept finding may of them every time I looked, even if I checked several times a day, but the last couple of weeks I have found them only once.

I found severe attack of leaf miners on spinach beet this year.  After pinching off damaged leaves (or part leaf if slight damage) removed the white eggs from undersides using a very small artist's paintbrush and the strongest alcohol available (I had 96%).  This dissolves the glue which sticks the eggs on so they lift off on the brush - and should kill any eggs not removed. 
If this fail, there is always the rest of the booze to finish!

PS In case a very small artist not available, a very small brush from a large artist will do.