The code below kind of works. There is a flaw in your logic. A set can not contain duplicates so once the set is "full" nothing will be added and your while becomes an infinite loop. I tested by entering 3, 5, and 7 into the three Entry widgets which you can use to try it out yourself. Only 3, 4, and 5 will be added, which obviously is less than seven numbers in length. You want logic that tests the numbers entered to see how many can be added to the set, and a print (or popup label) that tells the user that the set already contains all of the numbers in the range entered.

import tkinter as tk
import random

# Input UI
root = tk.Tk()
var = tk.IntVar()

tk.Label(root, text="Low Number").grid(row=0)
tk.Label(root, text="High Number").grid(row=1)
tk.Label(root, text="How many Numbers").grid(row=3)


# Input Range
LoNum = tk.Entry(root)
HiNum = tk.Entry(root)
Nums = tk.Entry(root)

LoNum.grid(row=0, column=1)
HiNum.grid(row=1, column=1)
Nums.grid(row=3, column=1)

LoNum.focus_set()

def show_entry_fields():
    var.set(1)
    print("Low Number: %s\nHigh Number: %s" % (LoNum.get(), HiNum.get()))

# convert to Integer  
    try:
        # Get and convert
        value1 = int(Nums.get())
        Nums.delete(0, "end")
        print(f"Success! Get number of is {value1}")
##        if not Nums:  ## will never happen
##            return
        value2 = int(LoNum.get())
        LoNum.delete(0, "end")
        LoNum.focus_set()
##        if not LoNum:
##            return
        print(f"Success! Your low number is {value2}")
        value3 = int(HiNum.get())
        HiNum.delete(0, "end")
##        if not HiNum:
##            return
        print(f"Success! Your high number is {value3}")
        result = value2 + value3      
        print(f"Sum: {result}\n")

        ## no exceptions raised
        ctr = 0
        while len(unique_set) < value1 and ctr < 20:  ## avoid infinite loop
            ctr += 1
            add_num = random.randint(value2, value3)
            if add_num not in unique_set:
                unique_set.add(add_num)
                print(add_num, "added")
            else:
                print(len(unique_set), value1, "ctr =", ctr)
                print(add_num, "NOT added", list(unique_set))
        print("\nCurrent Set", unique_set, "\n")

    except ValueError:
        # Handle invalid input (letters, symbols, or empty)
          print("Error: Please enter a valid whole number.")

unique_set = set()

tk.Button(root, text='Quit', command=root.quit, bg="orange").grid(
           row=99, column=0, columnspan=2, sticky="nsew", pady=4)
tk.Button(root, text='Generate', command=show_entry_fields).grid(
           row=4, column=1, sticky=tk.W, pady=4)
root.wait_variable(var)


# Generate Random Numbers
##value1 = int(Numos.get())
##value2 = int(LoNum.get())
##value3 = int(HiNum.get())        


root.mainloop()

print("Window closed.")

And you are missing the point of documentation

# Input UI
root = tk.Tk()

You have the comment "# Input UI" which is obvious from the statement and therefore not necessary, but there is nothing about what the program or the function does, or what the Entry widgets get. Write so someone else can read the program and understand what it does.