That looks like it is the same flush syphon as our two - a Dudley Duoflush. Holding the handle down, is intended to produce a short flush, a quick press should produce a full flush.
I've got my girlfriend to take a photo. Not sure if this helps at all, but at least you can see what I'm working with. She also did a video of it flushing, but I can't upload this.
How about using less paper, or flushing after using some paper then flushing when you have finished.
Tell her to stop putting wipes and tampons/sanitary towels down it.
Looking at the state of that cistern it's not been replaced in a long time and is probably due for an overhaul, also the water level looks like it can be increased to allow more water to flush down.
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