There is no estimate, because no one knows how many books were destroyed. There was no record of Qin's book burning before Sima Qian's Records of the Grand Historian, which was written almost a century after the fall of the Qin. Many historians believe Sima Qian might have purposely exaggerated the account to demonize the Qin and glorify the Han, which had abandoned Legalism in favor of Confucianism. With Sima Qian being the only source to mention this, we have to take what he said with a grain of salt.

According to The Records of the Grand Historian:

丞相李斯曰: 臣請史官非秦記皆燒之。非博士官所職，天下敢有藏詩、書、百家語者，悉詣守、尉雜燒之。有敢偶語詩書者棄市。以古非今者族。吏見知不舉者與同罪。令下三十日不燒，黥為城旦。所不去者，醫藥卜筮種樹之書。若欲有學法令，以吏為師。

Translation:

The Prime Minister Li Si said: Your servant proposes that all historians' records other than those of the Qin's be burned. With the exception of the court erudites whose duties includes possessing books, those [private individuals] under Heaven who possess copies of the The Classic of Poetry, The Classic of History, and the recorded writings of the Hundred Schools of Thoughts shall turn in these books to their governors or commandants for burning. Those who dare discuss The Classic of Poetry in public shall be executed. Anyone who uses history to criticize the present shall be executed along with his family. Officials who witness such violations but fails to report them shall be guilty of the same crime. Anyone who fails to burn the books after thirty days of this decree shall be tattooed and sent to do [hard labor] on the Great Wall. Books on medicine, divination, and forestry are exempt from burning. If there are those who wishes to learn the law, they shall visit their officials to study.

So according to Sima Qian, the Qin only burned books held by private individuals. The government archives still maintained copies of the books which were supposedly burned, and books on technology, medicine, etc. were spared. Interestingly, many of these books were lost not to Qin Shihuang but to the warlord Xiang Yu when he entered Xiangyang in the aftermath of the Qin's fall and torched the Qin palace (along with the archives) to the ground.