Dell Inspiron n5110 not charging battery

Yes, the battery is not an original part, but a compatible Green Cell. When I put it on, it showed a charge of 75% and was discharging normally. When I turn on the power, the battery does not charge. I uninstalled the drivers several times, with no effect. Now I charge it manually without the laptop and I will try what can be done with QuickSet.

" I already think I have а problem with the motherboard "

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No, it isn't a motherboard problem, it a third party battery that causing you problems. Please remove AC adapter, battery, press 'Power On' button for about 20 seconds to release Static Flea Power. Replace battery, power on system, hopefully third party battery charge to 100%.

Note: You might have to perform the steps one or two more times to get battery to charge

If battery still doesn't charge, run the windows troubleshooter for battery issues or uninstall / reinstall the battery driver in device manager.  You can view the steps in the link below:

How to Troubleshoot Dell Laptop Battery Issues | Dell US

Also, download / install the latest Quickset application from your Dell Product Support page:

Dell QuickSet Application | Driver Details | Dell US

A dell computer usually will disable a dying battery. You have to enable battery using the Quickset Application.

Once Quickset is installed, right click battery icon at the bottom of taskbar, you should be able to see the battery application.

If none of the troubleshooting steps fix the third party battery then it's time to ask the seller for a replacement or a refund. 

Dell Quickset / Battery Meter:

see this Dell Laptop Battery - Frequently Asked Questions | Dell US

The AC adapter is displayed correctly in the BIOS. He told me on the phone that it might be а defective battery, but he wanted me to send him the battery along with the laptop (because it's in another city) and then they would decide what to do, whether to replace the battery or repair my motherboard.

If the system recognizes the AC adapter, it WILL charge the battery -- unless the battery is either incompatible or faulty.

Give the seller a chance to replace the battery - if the replacement won't charge either, find another supplier.

Today I talked on the phone with the dealer from whom I bought the battery. After a 30 minute conversation, he told me that my motherboard was damaged. Can I somehow check if it supplies power to the battery and how much it should be? The battery port is 9 pins.

Now I ordered an original battery Dell and it will be with me on Tuesday.

I only bought the battery from this seller. He is trying to tell me that the problem is not with the battery, but with the motherboard. So I wanted to know if there was any way to know if the motherboard sent current to the battery.

The system is smart enough to detect both the battery and AC adapter.  If the AC adapter is recognized and the system shows the battery charging, the charging system is working correctly.  That leaves a faulty or incompatible battery as the prime suspect if it's not actually taking a charge.

Carefully check the date code on the new battery -- this system is about a decade old, and rarely are new OEM  batteries available for more than five years, so you may well find your "new" battery is relatively aged before you start using it.