The success rate of a dynasty can be measured on the basis of various parameters like territorial expansion, the stability of succession lines etc. So, let us take a look at all the 5 Delhi Sultanates and then conclude our answer:

• Slave Dynasty (Mamluks): The success rate of this dynasty was high because after getting an entrance to the Gangetic plains, it was quite easy for these invaders to attack all the Gangetic plain kingdoms riding their mobile horses. The Indian kings did not get any time to replan their strategies as they were not accustomed to the fast-moving Turkish horses. Hence as you can see, they gained control of the entire Gangetic plains till Bengal in just a decade. However, after that, they were unable to encroach any further. So, the large territorial expansion was only due to the inexperience of Indian kings in dealing with the mobile Turks. However, they had a strong line of rulers compared to the other sultanates and thus they were able to survive for around 8 decades.

• Khalji dynasty: They can be considered as the most successful dynasty on the basis of the sheer size of its expansion under Alauddin Khilji. With his brutal and cunning strategies, he proved to be a successful strategist. During his time, South had already weakened due to the various interstate conflicts. Cholas had demised and Pandyas had been weakened. This fact was utilized by Khilji to acquire the never raided treasuries of the South. These vast treasuries helped him to finance his campaigns in North India. However, the Khiljis practically ended with the demise of Alauddin Khilji. So, one can say that Alauddin was unable to create a stable line of succession.

• Tughlaq dynasty: Though under the Tughlaqs, Delhi Sultanate was at its zenith, I don’t consider it as the most successful dynasty. Most of this expansion happened under Muhammad Bin Tughlaq, however, it was because of his policies that the demise of the Sultanate started. He overstretched his resources and finally ended up becoming a captive of Rana Hammir (Ancestor of Maharana Pratap). The period marked the entry of Ranas of Mewar in the power politics of North India.

• Sayyid: The weakest of all the Delhi Sultanates, they came to power after Timur ended the Tughlaq dynasty, after sacking Delhi. They were unable to survive for even 4 decades.
• Lodis: Though the Lodis under Sikandar Shah were able to regain some of the lost territories under the Sayyids, they were unable to curb the rise of new powers like the Ranas of Mewar (Ankit Pandey's answer to How did Rana Sanga break the Sultanate’s hegemony in North India?). Finally, their rule ended with the defeat in the First Battle of Panipat.

So let’s rank them:

On the basis of the increase in empire size:

1. Khilji
2. Mamluk
3. Tughlaq (Southern states were already feudatories of Khilji. They were just annexed by Tughlaqs)
4. Lodi
5. Sayyid

On the basis of the time the dynasty survived:

1. Tughlaq
2. Mamluk
3. Lodi
4. Sayyid
5. Khilji

So, according to me, the Mamluks were the most successful.

Probable Reason: Mamluk sultans were generally the most prominent slaves who had the capabilities of a strong leader. The line of succession was not determined by blood relations but by the survival of the fittest and hence Mamluk dynasty was more successful than the other Delhi sultanates.