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Battery change does nothing - do you have any more info on the solder fix? I've no experience soldering but my step dad might be able to help...
gillies-
Basically, they usually switch on when you put a battery in (provided the battery isn’t flat). If it switches on when you insert a new battery, but not when you press the on button, it suggests the switch has broken. If it appears completely dead even with a working battery I’m not sure what it could be! The scales are easy enough to dismantle with a Phillips screwdriver and a set of mini screwdrivers. Having a look at the circuit board might give you an indication as to what the problem is. Look for discolouration or corrosion. It’s harder to tell if a switch is knackered without destructively dismantling it.
Replacing the switch is a simple enough affair. Remove the solder on the switch using a soldering iron and a solder sucker of some sort, then bang a new momentary switch on.
I used these, 6x6x4.3mm
https://www.ebay.co.uk/itm/231957884472
Nef
If you take the battery out, then reinsert, do the scales come back on?
It may be that the power switch has broken.
These models both have switches that easily water damage. An easy fix with a soldering iron and a new momentary switch (10p). I repaired a pair I had this way.