Disclaimer: In KL for research atm so don't have many books to hand as a result this answer will be a bit schematic sorry.

The impact of the sack of Baghdad on the Islamic world is rather difficult to quantify as the significance entirely depends on the angle you examine it from. In purely political terms the consequences of the fall itself were somewhat limited. The Abbasid caliphs' power had long been waning they controlled little territory in reality with most of their theoretical domain dominated by independently powerful and squabbling governors. The more active political centres of the Islamic world had move west to places like Cairo.

On a more cultural level a much vaunted, but very flawed, theory sees the sack of Baghdad as the start of some sort nebulous decline of Arab civilsation. This is often coupled with references to the equally dubious story about the destruction of the famous Abbasid library they House of Wisdom, the Mongols supposedly throwing so many books into the Tigris it ran black with ink. This is actually almost certainly false the claim being written by scholars who were nowhere near Baghdad at the time but had ample reason to hate the Mongols. In fact Hulagu, the Mongol general, had as a vizier the famous astronomer Nasir-al-Din al-Tusi who reportedly saved some 400,000 manuscripts. For the Islamic world more generally George Saliba comments that the 13th century was in fact a particularly fertile time for Islamic science. He even goes so far as to argue that the destruction of Baghdad may have helped the process by energising other centres of scholarship like Damascus, Cairo and Isfahan. Equally the Mongol Ilkhans, who would later convert to Islam, proved major patrons of the arts, sciences and trade.

However, we should not underestimate the psychological impact on the Sunni Islamic world. While the Caliph had not wielded much practical power for a long time the office remained of massive symbolic importance. Thus to see it snuffed out was seen as a terrible sign of things to come and made the Mongols appear especially grotesque and terrifying. Returning to the allegations of throwing the books into the river while we may see in retrospect how the impact was perhaps limited claims such as this shows how the fall of Baghdad was interpreted as a massive blow by the barbarism of dar al-harb against the civilisation of dar-al islam.