I've only got limited experience of laptops. My neighbour gave me an old Dell XP thing he would have otherwise thrown away. I didn't keep it long because my sister needed something with Excel on so I lent it to her. Needless to say, I bet I won't see it again.
Anyway, could it be the socket in the laptop where the charging plug goes in ? I notice you said the new battery lasted a week, so I'd guess it may have taken some charge during that time, because I'm sure even a fully charged battery would struggle to power a laptop for a whole week without getting any charge at all. I can't remember if mine had a charging light on the laptop itself. I know it had a green one on the charger.
It would only take a dry solder joint on the socket in the back of the laptop to cause an intermittent failure, but without knowing the intricacies of your actual computer, I don't know how you're meant to tell if it's getting a charge or not. I suppose you could start by carefully testing the output of the charger using a voltmeter. Assuming you get a decent reading there, then maybe you could remove the battery and check the terminals it connects to, with the charger plugged in and powered up, to give you an idea if the charge is getting into the laptop itself.
that's certainly a possibility, or an incorrect charger.
they would typically be either 12v or 19v, but some chargers output 19.5v, again if using a wrong type,then the PC would read this as over/under voltage and shut off the charging mechanism.
Also the polarity of the charger should be checked.most are outside ring "earth",but you do get the odd type where that polarity is reversed.
a bad joint or connection is also another possibility.
a voltmeter would confirm voltage to the end of the PSU cable, but not its performance under load.
you would probably need some sort of electronic schematic to check the laptop pc connector though, by setting the voltmeter to resistance and running a continuity check you could ascertain if the charging circuit was operating correctly or if something like a capacitor ,regulator or varistor had blown.
or an open circuit(infinite/very high resistance )if you have a dodgy connector.
The only other way would be open the laptop up and visually inspect the solder joints.
first port of call I would try substituting the psu and battery for known genuine working parts before tearing anyting to bits....if you know anyone with a similar laptop design that you can borrow the bits for a while, do that first, then scout ebay or gumtree for any parts that might be floating around on there.
considering an new psu or battery is likely to set you back £20-30 each , you really need to keep an eye on cost. you can buy a far more powerful generic chromebook type thing for less that £100 new these days, and you can also buy ex business hand me downs(core i3/i5) for about £50 on auction sites, so if you can't fix for less than that, then it's time to buy another one..
