How to increment %eax register and output in a c callable function?

Question

#include <stdio.h>
extern int count(char *string, char c);

int main(void)
{
  char s[100];
  char c;
  printf("Enter a string of characters:\n");
  scanf("%s", s);
  printf("Enter a character to count:\n");
  scanf("%c", &c);
  printf("\nThe number of %c's in the string %s is %d\n", c, s, count(s, 
c));
  return 0;
}

I'm trying to run this code that will count how many char c's are in the string s through an external function in the x86 assembly below, but when I debug it in gdb, it gives me 0 no matter what and when I set a breakpoint before at the count function, it just skips over it and completes the program returning 0. I've tried debugging this for a good amount of time but I just can't figure it out, so does anyone have any pointers or suggestions that could lead me in the right direction? I'm new to assembly so I'm not sure if any of the code below is correct or if anything is unnecessary. Any help is greatly appreciated thanks!

.globl count
.text
# first argument is string, second argument is character

count:  
    push %ebp       #push old ebp to top of stack
    movl %esp, %ebp     #point ebp to same thing as esp, both pointing to old ebp

    #saving live registers
    push %esi
    push %edi
    push %ebx   

    #take values from stack
    movl $0, %eax       #set count(eax) to 0
    movl 12(%ebp), %ecx     #move letter to be counted to ecx
    movl 8(%ebp), %esi  #move string to edx

lettercheck:
    cmpb (%esi), %cl    #compare current letter to ecx(letter to be counted)
    incl %esi       #next letter
    je inccount     #if match inccount 
    cmpb $0 ,(%esi)     #if no match and end of string
    je end
    jmp lettercheck     #if not end of string and no match  

inccount:
    incl %eax       #inc count of char
    cmpb $0, (%esi)     #check for end of string
    jne lettercheck     #if match and not EOS lettercheck
    jmp end         #else jmp end
end:
    pop %ebx
    pop %edi
    pop %esi
    movl %ebp, %esp
    pop %ebp
    ret