Well, when I connect it with computer via USB then Arduino IDE does not detect it.
Again, we need more intricate detail. You have a Nano board, and you have a (slightly unusual) "IO shield". You can connect the Nano to the computer without it inserted and you can connect the Nano to the computer while inserted in the "shield". So which is it? Have you tried the Nano on its own and does the IDE detect it? If not, then you clearly have a problem which is nothing to do with the shield.
And needless to say, it is the Nano which you are connecting to the computer, not the shield, and while it is connected to the USB, it will be powered by the USB - the pilot LED on the Nano should be lit. While this particular shield has no pilot LED, all the basic Nano expansion shields do.
Left down corner in this image
Yes, that is where the 2.1 mm standard "barrel jack" is mounted. You can get connector leads for this on eBay (and more quickly at greater expense, various electronics parts suppliers). See below.
However when I'm searching internet I also find things like this
and here + 11V is connected to VIN and - to GND. GND can see on shield but VIN not.
Well, that is the Nano itself, not the shield.
What I'm missing? And is there any easy way of connecting it? Buying some pins or something? Soldering battery doesn't sound like a good idea. I hoped that with Nano and proper cables I can have it solderless.
If you intend to use this shield without soldering to it, get the corresponding lead.
I just now also found that I have 2 GND on Arduino Nano. Does it matter how I connect it (which I use for battery and which for GND for other electronics?
Mostly not, but the connections to the ground pins should be short. Note on your shield, I count no fewer than 26 ground pins! :grinning: