What is the ruling on the prayers performed by one who only wiped over the sling of a splint, thinking that this was the correct way of wiping, and then came to know that wiping should be over the splint itself?

If you were wiping over a string on which the splinted hand was hanging, and not over the splint itself, then your ablution is invalid and the prayers must be repeated. If the "sling" was part of the splint and you did not wipe over its entirety, then your ablution is invalid according to the majority of scholars, because it is obligatory to wipe over the entire splint that covers the obligatory area. Some scholars have held that wiping over most of the splint is sufficient. Due to the differing opinions of scholars, it is more prudent to repeat the prayers performed with this ablution. If you do not know the number of missed prayers, some scholars have said to pray until you are reasonably certain that your obligation has been discharged (this is the view of the Hanafis and Malikis, and some contemporary scholars), while others have said to pray until you are certain of the discharge of your obligation (this is the view of the Malikis, Shafi'is, and Hanbalis). So strive to make up what you are reasonably certain will discharge your obligation.