How can I repair a Nintendo handheld device if there isn't a service center in my country?

As a hobby, I refurbish old Nintendo handhelds and I've even replaced (cough, multiple, cough) screens for my wife's cell phones over the years. I can't speak for the 2DS XL, but replacing internal components can range from "easy" to "killer, expert mode!" I joke a little, but it truly is amazing how complex some components sometimes fit together in electronic devices but if you get lucky, some are simply placed within a housing assembly and can be as easy to change as unplugging a wire, and pulling out and snapping in a new component. That's getting to be rarer and rarer, but it does still happen from time to time.

Many people have repaired their own gaming hardware and had good success. However, if you feel that buying the components and fixing the 2DS yourself is to great a risk, I completely understand. Considering your circumstances I'd recommend you check around your town, or the nearest one, and see if you can find an independent electronics shop. Maybe a place that sells televisions or appliances. Go there, ask to speak to a manager and tell them your problem. They probably do not offer in-house repair, but if it's an independent shop and you can speak to the head manager or the owner, there is a good chance that they may be connected with people in the community who have done electronics repair.

I can't speak for Latvia, but living in the US, electronics repair is a dying art but some of us enjoy keeping it up as a hobby. Tech is often too disposable for us today, so the need for the skill is dying. But, there are still people who know how to fix things and even if they are retired, they might enjoy taking on a project like this just for the experience.

If you can get a local contact for someone who has done electronics repair, give them a call and see if they are willing to give this repair a try. Let them know you'll order the components and once they arrive, you'll deliver everything they need for the repair, and all they will have to do is the replacement work.

I know this isn't a great answer, but it sounds to me that your real problem is finding someone capable of doing the work. If Nintendo repairing the item isn't an option, then you will have to find someone who can. People who do this as a side-gig or as a hobby are rare, but there's a good chance that more than a few people in your town were possibly once electronic repair technicians and those potentially older, retired individuals still tend to enjoy dabbling in the field when they get a chance.

As a last resort, you could try to go to an independent cell phone repair shop. Again, I say find an independent one because if it's a chain store they probably won't be allowed to help you with specialty hardware. However, if there is an independent shop and you walk in and show them your system and tell them you're willing to buy the hardware and deliver it for the repairs, they may consider doing such a repair. Even if you can't find an independent shop, maybe, if you can talk to the local repair technician, you might be able to persuade him to do this repair as a side-gig.

I really don't see there being any other options but, honestly, I don't think you're completely without a solution. It may take a bit of searching and asking around in your local community to find the right person, but I'm sure they are out there.