After cleaning the stove holes with some liquid detergent, one of them is clogged now. I used a sewing needle, without any effect. Did anybody run into this problem before?
The jets can be unscrewed, using a properly sized wrench, and cleaned with an appropriately sized cleaner. Once clean they can screwed back in.
They are typically made from brass and are soft. Using the wrong wrench can round off the hex head. More importantly the orifice in the jet can be unintentionally enlarged or damaged easily. Inserting anything from the outside will push debris into the fuel system and while it may clear the hole it leaves the debris inside. It is better to remove them for cleaning. If damaged they are easily replaceable. Before doing anything check the documentation for your stove. The installation or maintenance instructions may include directions for replacing the jets.
I use an oxy-acetylene torch tip cleaner on my stove jets if they are clogged. My tip cleaner provides sizes 6 to 26 one of which fits my jets.
To clean them I select the largest cleaner which easily fits into the orifice. It is inserted and gently run in and out to clean the orifice. Rather than being smooth the cleaners have very small ridges on their sides that help to remove debris. If not used carefully they can act like a file and enlarge the size of the orifice in the jet. Keep the cleaner aligned with the orifice vice at an angle. It needs to go straight in and out. The cleaners can also be bent when inserting. Hold them close to the insertion point when clearing the jet. If it seems too hard try a smaller size cleaner.
I do not turn off my gas supply, just insure the burner is off and stays that way until you are done.
You could try heat.
Both jets contribute to the manifold. You could light it and let it run and hope that the heat will unclog the blocked one.
Or you could leave the manifold off and heat up the jet with a flame - maybe just from a lighter, or a cool flame from a torch. If it is moisture in there that could dry it up.