Python, 74 64 56 55 bytes

x=input()
 a=ord(max(x))-54
a+=7*(a<8)
print(int(x,a),a)

This is case-sensitive, requiring upper case letters, and crashes for invalid input (e.g. "$^$%!(&£%)())

This code gets the max letter in the string (z>x>y>...>b>a>9>8>...>2>1>0) and gets the largest base needed to represent it.

Change the 54 into a 76 to make it work for lowercase (but no longer upper case).

Python 2, 74 64 56 55 53 bytes

def f(x):a=ord(max(x))-54;a+=7*(a<8);print int(x,a),a

Call the function (i.e. f("HELLOWORLD")) and outputs to stdout (809608041709942 33)

This is case-sensitive, requiring upper case letters, and crashes for invalid input (e.g. "$^$%!(&£%)())

This code gets the max letter in the string (z>x>y>...>b>a>9>8>...>2>1>0) and gets the largest base needed to represent it.

Change the 54 into a 76 to make it work for lowercase (but no longer upper case).

Python, 74 64 56 bytes

x=input()
a=ord(max(x))-54
if a<8:a+=7
print(int(x,a),a)

This is case-sensitive, requiring upper case letters, and crashes for invalid input (e.g. "$^$%!(&£%)())

This code gets the max letter in the string (z>x>y>...>b>a>9>8>...>2>1>0) and gets the largest base needed to represent it.

Change the 54 into a 76 to make it work for lowercase as well.

Python, 74 64 56 55 bytes

x=input()
a=ord(max(x))-54
a+=7*(a<8)
print(int(x,a),a)

This is case-sensitive, requiring upper case letters, and crashes for invalid input (e.g. "$^$%!(&£%)())

This code gets the max letter in the string (z>x>y>...>b>a>9>8>...>2>1>0) and gets the largest base needed to represent it.

Change the 54 into a 76 to make it work for lowercase (but no longer upper case).

Python, 74 64 bytes

x,i=input(),2
while 1:
 try:print(int(x,i),i);break
 except:i+=1

Alternative 64 bytes

x=input();i=z=2
while z:
 try:print(int(x,i),i);z=0
 except:i+=1

This is case-insensitive, and loops forever for invalid input (e.g. "$^$%!(&£%)())

This code uses Pythons built-in int to try and convert the string into base 10. It starts at base 2 and tries all. If there is an error (It is not base i), it continues after incrementing i. It does this until it successfully prints the number.

Python, 74 64 56 bytes

x=input()
a=ord(max(x))-54
if a<8:a+=7
print(int(x,a),a)

This is case-sensitive, requiring upper case letters, and crashes for invalid input (e.g. "$^$%!(&£%)())

This code gets the max letter in the string (z>x>y>...>b>a>9>8>...>2>1>0) and gets the largest base needed to represent it.

Change the 54 into a 76 to make it work for lowercase as well.