Thank you, Rabbi, for the answer.
In our case there were many small children.
Does that count as a great need, or should we simply have eaten cold food and let the refrigerator spoil?

The benefit that the non-Jew who performs the act needs to have must be direct benefit—for example, food that he needs would not spoil if he raises the refrigerator fuse, and so on. The orange juice he received afterward is not considered benefit, because that kind of benefit always exists—whether in the form of a bottle of orange juice or in the form of gratitude.

It definitely seems to me like a great need. The source that permits instructing in the case of a Torah-level prohibition deals with lighting a lamp for illumination, which seems far less essential.

Rabbi, why is eating hot food a great need? People eat cold food every single day. You can eat cold and manage fine even if hot food is considered tastier, no?
A lamp for light at night, at least in their time, really was more necessary because you couldn't see at night (there was no urban night lighting outside and so on that could cast some light into the house). It may be that nowadays this is actually a less serious need (depending of course on the location, etc.).

First of all, the refrigerator also went off. Second, nowadays cooked food that isn't hot is definitely a significant need, certainly for children. In their days they lived in darkness, so in my opinion the need for a lamp really wasn't all that acute. There was natural light from the moon and stars and other people's candles. If I think about us nowadays, it seems to me that the overwhelming majority would prefer darkness during the meal over having no hot plate and no refrigerator.

Assuming the problem had only been with the hot plate (say the refrigerator had been connected to a different power source), would you still think so?
I don't know, Rabbi, I think children too could manage for one Sabbath with lukewarm food. Like eating a sandwich, or cold schnitzel, or cold pizza. It's not the end of the world, especially if it's a one-time case.

If the problem had only been the hot plate, there would be room to discuss it. It depends what food there is and what the children eat. I don't have a general answer.

According to most halakhic decisors, who hold that turning on electricity is rabbinic, there is no problem here (except that according to Rabbi Michi it is Torah-level).
However, the hot plate is Torah-level, but her intention in raising the breaker is not for the hot plate but for the general electricity, so regarding the hot plate this is an inevitable consequence, which is permitted in instructing a non-Jew.

Actually, I had a disagreement with my friend about the incident that happened to us.
Originally we went out into the street to look for someone who had a Sabbath hot plate, and we found a nice woman who offered to let us use hers.
My friend asked her whether she knew a foreign worker, and she said yes.
I thought it would be better to use that woman's hot plate and not ask the foreign worker to raise the breaker. As was mentioned here, the foreign worker received no direct compensation for raising the breaker—we only gave her orange juice indirectly.
My question is:
If in such a case we have an alternative for heating the food, or even the option of eating it cold, is it still permitted to ask the non-Jewish woman to raise the breaker by hinting?
And is the very act of asking a non-Jew by hint to desecrate the Sabbath considered a Torah prohibition or a rabbinic one?

The Rabbi noted that he does not distinguish between hinting and explicit instruction to a non-Jew.
But according to the content of this video

there is a difference if I understood the video correctly. The Rabbi there (in the context of hinting to a non-Jew regarding a permitted act) seems to imply that it is permitted to direct a non-Jew, for the sake of Sabbath heating for example, to turn on a hot plate in order to warm food for small children—and that we too may benefit from the hot food.

It is a rabbinic prohibition. This is a case of instructing regarding a Torah prohibition (in my view), and therefore one should not be lenient except in a case of great need. In the case of instructing regarding a rabbinic prohibition, they permitted it for the sake of a commandment, and Sabbath enjoyment is a commandment. The Rema wrote that there is no difference between explicit instruction and hinting, and that is the view of most halakhic decisors. The Bach, however, distinguishes between them.

Thank you, Rabbi, for the response. I would also be glad to hear the Rabbi's opinion on the situation as it actually was, and on my previous question.
My question is:
If in such a case we have an alternative for heating the food, or even the option of eating it cold, is it still permitted to ask the non-Jewish woman to raise the breaker by hinting?
And is the very act of asking a non-Jew by hint to desecrate the Sabbath considered a Torah prohibition or a rabbinic one?

I answered everything. Instructing a non-Jew is a rabbinic prohibition. Simply speaking, there is no difference between explicit instruction and hinting.
There is a difference between instructing a non-Jew to do a rabbinic prohibition (a rabbinic prohibition upon a rabbinic prohibition) and instructing him to do a Torah prohibition (a rabbinic prohibition involving a Torah prohibition).
A rabbinic prohibition upon a rabbinic prohibition is permitted for the sake of a commandment (that is, for a commandment-related need). A rabbinic prohibition involving a Torah prohibition is customarily permitted only for a very great need.
Turning on a hot plate is, in my view, a Torah prohibition, but many halakhic decisors hold that it is only rabbinic.
A need for hot food is a commandment-related need (Sabbath enjoyment), but not a very great need.
It follows that in my view this is a rabbinic prohibition involving a Torah prohibition for the sake of a commandment (and not a very great need), and therefore it is forbidden. But according to many halakhic decisors this is a rabbinic prohibition upon a rabbinic prohibition, and it is permitted for the sake of a commandment.