You informed the landlord that you wanted to terminate the lease early.
The landlord offered to agree to the early termination in exchange for payment of 2 additional months' rent beyond the time that you occupied the premises. You agreed and fully performed under the agreement. Whatever happened after that with the landlord and the premises is none of your concern.
At the time you made the agreement, the landlord knew (or should have known) that the possibility existed that he/she/it might not find a new tenant to occupy the premises before 2 months after you vacated. In order to account for that possibility, the landlord
could have made you a different offer: that you would pay a minimum of 2 additional months' rent, up to rent through June, depending on when the landlord found a new tenant through the use of reasonable diligence. The landlord did not do that and took the risk that it might not find a new tenant within the 2 months.
At the time you made the agreement, you knew (or should have known) that the possibility existed that the landlord might find a new tenant to occupy the premises before 2 months after you vacated. You
could have countered the landlord's offer by offering to pay 2 additional months' rent,
but, if the landlord found a new tenant within the 2 months, you'd get some of that additional rent. You did not do that and took the risk that the landlord might find a new tenant within the 2 months.
Each of you and landlord took a risk when making the agreement to terminate early. You paid a fee to get out of the lease early and considered that fee to be reasonable. Now you want to say, "although I considered our agreement to be fair, I'd now like to alter the agreement, after the fact, because we could have made a different agreement that would have been more favorable to me." Do you really think that's reasonable?
Let me ask you this: if the landlord was unable to find a new tenant within the 2 months period and came back to you asking for more money, would you be willing to pay more?
If it is not in your lease I can't imagine why you ever paid it.
The OP paid it because the alternative was to be stuck with a lease through June. The "fee" gave the OP certainly with respect to how much of the rent he/she would be stuck with after vacating.