I would put small blocks under the ends at each side. Screw these to the strings, (the sloping sides), and then screw down through the steps into the blocks.
Make the blocks about 38mm x 38mm and the length of the step from back to front. Clamp them under the step using 2 G-clamps and then drill a 3mm pilot hole right the way through the block and the string. Using screws long enough to go through the string, and half way into the block, screw in from the outside edge all the way through into the block. So, if the string is 38mm thick and the block is 38mm thick then 65-70mm screws, (maybe 4 or 5mm diameter), will do the job. Use countersink ones so they don't protrude and catch little fingers or legs.