Let's do a quick test, unplug the battery then plug the charger in.
If you get an apple logo on and off, it looks like the charging IC is probably okay.
If you do not get apple logo on and off, charging IC needs to be replaced, as it can't charge batteries under 3.7v or a.k.a flat batteries.
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The U2 Tristar Charging IC can only be replaced by people that do micro soldering. You may be able to find one in your area that does this service, for example mobile phone repair shops.
Let's do a quick test, unplug the battery then plug the charger in.
If you get an apple logo on and off, it looks like the charging IC is probably okay.
If you do not get apple logo on and off, charging IC needs to be replaced, as it can't charge batteries under 3.7v or a.k.a flat batteries.