In situations like this, the hard drive will continue to spin, (and the Ethernet ports will continue to light up), but the failing power supply inside the Time Capsule cannot provide enough current to run the router portion of the Time Capsule.
Apple never designed the Time Capsule for service, so they have not ever provided repair parts for the Time Capsule. Even if a replacement power supply could be located and removed from another Time Capsule, it would not make sense to attempt repairs on the product unless you also were planning to replace the internal hard drive. At 11+ years old, it is already 3-4 years past due to fail.
Frankly, it is remarkable that the Time Capsule has gone as long as it has, since most fail between 5-7 years. Congrats on your very good luck with the product.
Repairing the power supply and replacing the hard drive......if you can find a shop to do this......is going to cost a good deal more than the original cost of the Time Capsule.
If you value the data on the Time Capsule's hard drive, my suggestion would be to open up the Time Capsule and remove the hard drive, then place the hard drive in a caddy or enclosure so you can connect it directly to the USB port on your Mac. If you can "see" the contents on the old hard drive, copy all the data over to another new hard drive.
Take the old Time Capsule to a recycling center, but keep the old hard drive in your desk drawer.