Is her slaughter halal, as she slaughtered a pigeon with a knife that was not sharp, and only a little blood flowed?

Firstly:

The conditions of slaughter being halal

In order  for the slaughtered animal to be halal, it should be cut in the right place, so the two veins should be cut, along with the trachea – which is the windpipe – or the oesophagus – which is the food pipe – ; or one of the veins should be cut along with the trachea and oesophagus.

Ibn Qudamah (may Allah have mercy on him) said in al-Mughni, 9/316: There is no difference of scholarly opinion regarding the fact that the best way is to cut all four: the trachea, the oesophagus and the two veins. The trachea is the windpipe and the oesophagus is the food pipe, and the two veins are those on either side of the trachea. This is the best way because it is the quickest way to dispatch the animal, because it causes the least pain and avoids an area of scholarly dispute, so it is more appropriate. End quote.

He (may Allah have mercy on him) mentioned that the view of ash-Shafa`i and Ahmad (may Allah have mercy on them both) is that it is stipulated that only the trachea and oesophagus be cut. The view of Malik is that it is stipulated that all four be cut.

The view of Abu Hanifah is that it is stipulated that the trachea and oesophagus, and one of the veins, be cut.

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: The oesophagus, trachea and the two veins must be cut. The strongest view is that cutting three out of the four is permissible, whether that includes the trachea or not, because cutting the two veins is more effective than cutting the trachea, and it is more effective in making the blood flow.

End quote from al-Ikhtiyarat, p. 468.

If three out of the four were cut, then the bird you slaughtered was halal, and it does not matter if the blood that came out was a small amount, or the bird did not move after it was slaughtered.

But if you did not cut anything except one vein only, then it was not halal.

Secondly:

It is disliked (makruh) to slaughter with a blade that is not sharp

It is disliked to slaughter an animal with a blunt instrument, because that causes unnecessary suffering to the animal, and because Muslim (1955) narrated that Shaddaad ibn Aws (may Allah be pleased with him) said: There are two things that I memorized from the Messenger of Allah (blessings and peace of Allah be upon him). He said: “Allah has prescribed kindness in all things, so if you kill, kill well, and if you slaughter, slaughter well. Let one of you sharpen his blade and spare suffering to the animal he slaughters.”

It says in Kashshaf al-Qina`, 6/210: And it is disliked to use a blunt instrument, because it causes unnecessary suffering to the animal. End quote.

Shaykh Ibn `Uthaymin (may Allah have mercy on him) said: “Let one of you sharpen his blade” – this is a command instructing him to make it sharp so that it will cut quickly. The blade is the knife.

What is meant is: if you want to slaughter an animal, slaughter it with a sharpened knife, as that is more likely to cut without causing pain.

“and [let him] spare suffering to the animal he slaughters” – this is an additional instruction, in addition to sharpening the blade. What is meant is to cut forcefully, placing the knife on the neck and cutting it with force, so that will be quicker than cutting two or three times. There are some people whom Allah enables to cut the two veins, the trachea and the oesophagus in one go, because they hold the knife firmly and the knife is good and sharp, which makes the animal die without suffering.

End quote from Sharh Riyadh as-Salihin, 3/598.

He also said: We learn from this hadith that it is obligatory to sharpen the knife, because he said “Let one of you sharpen his blade.” If we say that sharpening the blade is obligatory, that means that slaughtering with a knife that is not sharp is prohibited.

But is the animal that is slaughtered halal?

Yes, it is halal, even though this action is prohibited, because the Prophet’s words “Whenever the blood is caused to flow and the name of Allah is mentioned over it [the animal that is slaughtered], then eat” are applicable in this case.

We also learn from this hadith that it is obligatory to spare suffering to the animal being slaughtered, because he said, “and [let him] spare suffering to the animal he slaughters.” So he should use the method that is most likely to spare suffering to the animal, because the basic principle is that causing pain to the animal is prohibited, but Allah, may He be Glorified and Exalted, has permitted doing so to serve people’s interests. Based on that, we should reduce the suffering to the minimum that is required when slaughtering, and spare unnecessary suffering to the animal on the basis of what we have said above, that the knife should be sharp and that slaughter should be done firmly and swiftly.

End quote from Fat-h Dhi al-Jalal wa’l-Ikram, 6/64.

So you must seek Allah’s forgiveness for slaughtering this bird with the blunt knife, and not do such a thing again.

And Allah knows best.