Did Keane throw away a chance of destroying Jackson after his failed night attack ? https://www.google.com/amp/s/www.history.com/.amp/topics/war-of-1812/battle-of-new-orleans
As a footnote to the Battle of New Orleans, it was the first artillery battle in the US and it was the US artillery that caused the majority of the British casualties.
Regarding the American night attack of 23 December 1814 at New Orleans against Keane, the following might be of some help:
From Amateurs, To Arms by John Elting, 299:
'It had been a lively affair. Had Jackson had with him the regular regiments he left in useless idleness at Mobile, it could have been a decided American victory. He simply expected too much from utterly raw militia. Even so he had won a psychological victory. Casualties were not far from even-American: 24 killed, 115 wounded, 74 missing; British: 46 killed, 167 wounded, 64 missing. The suddenness and fury of Jackson's assault, however, convinced Keane that he had been assailed by 5,000 Americans. The Carolina, which Patterson had shifted over to the west bank, kept firing at any movement in the British camp. Giving up all thoughts of an immediate advance, the British concentrated on getting troops, guns, and supplies ashore.'
From The US Army in the War of 1812, Volume II, 852-853:
'The British certainly did win a tactical victory, which enabled them to maintain their position. The main credit is due to Captain Hallen (Captain William Hallen, mentioned by Keane in his report: 'A most vigorous attack was then made on the advanced front and right picquets, the former of the 95th under captain Hallen, the latter the 85th under captain Schne; these officers and their respective picquets, conducted themselves with firmness, and checked the enemy for a considerable time...'). Had the Americans driven him from his post in their early attack, they could have inflicted a major disaster on Keane's forces. Hallen gave the British time to get some order established and also for British reinforcements to come up. These played a part in diverting Coffee from his attack. Secondarily, credit goes to Colonel Thornton who remained unflappable and with his sturdy defense kept Coffee from driving the British into the river. However, that claim to a complete victory is not justified. Gleig and Subaltern both speak of the Americans as being driven in greatest discord from the field. There was no rout, and the Americans were not driven from the field. They withdrew themselves and remained near it until 4:00 AM.'
'The 'Night Battle' was, however, a moral victory for the Americans. It is not too much to say that it saved New Orleans. The British were disabused of their expectation of an easy conquest. The unexpected and severe attack made Keane even more cautious. Even though he was constantly reinforced until his whole army was concentrated on the field, he made no effort to advance on the twenty-fourth or twenty-fifth. Instead, his troops remained cold and miserable between the old and new levees sheltered from Carolina's guns and those of Louisiana which joined her on the twenty-fifth.'
From The British at the Gates by Robin Reilly, 258:
'Keane's army was in no state to attempt pursuit. He had lost 276 officers and men, including 46 killed and 64 missing, and his camp was a shambles of dead and wounded.'
So the answer to the OP question is no. Keane was in no shape to follow up his tactical triumph and the Americans had withdrawn. They had not been routed. The British had been surprised and had lost more than the attacking Americans and also believed that the Americans had fielded more troops in the attack than they had.
Keane had the same problem with enemy estimations of strength that both Riall and Drummond had in the Niagara battles in 1814. All three thought they were facing more Americans than they actually were.