Short answers, pulled from the story.
Jean Reynier commanded a force of 2,160 soldiers under Napoleon's broader offensive against the Ottoman Empire. Louis Caffarelli started engineering work before the French began artillery fire on the fortifications.
The French successfully captured El Arish from the Ottomans on the 20th of February 1799 after nine days of intense fighting. The siege officially began when Kléber's division reached El Arish on the 12th of February 1799.
The French had lost 200 killed and 300 wounded during the entire campaign against the fortress. They also lost only 3 men during an initial engagement where they captured all twelve Ottoman cannons.
The Ottomans suffered about 500 killed along with 900 prisoners of war captured by the French. Around 300 of those switched sides and joined the French army while forty escaped to settle in Great Britain.
The capture of El Arish secured the northern flank of the French occupation in Egypt for months to come. This successful operation demonstrated Napoleon's ability to project power deep into Ottoman territory despite logistical challenges.