You need to show that $k+1 = 1$ on the basis of the inductive hypothesis that $k=1$ for some arbitrary $k$. So, you need to show that $1+1=2$ .... but you do no such thing. You just state what you need to prove, and apparently take that statement as proof.
The moral: Don't just write down a bunch of mathematical expressions when trying to prove something. Always clearly separate between what you assume and what you are trying to prove.
Now that I understand what you're trying to do and what your question is ... Allow me to first rephrase your question. You are basically asking:
Suppose I believe that all numbers are equal to $1$. Then it doesn't seem like induction will help me prove that I am wrong about this, because when in the inductive proof I get to the point where I end up showing that $2=1$, I don't get a contradiction with my belief, since my belief says exactly that: $2 = 1$, yes, that's right! More general then, it seems like induction cannot show that you are wrong about something since the result of the induction will be compatible with your belief. Is this true?
Answer: No. As long as you believe certain other things, you can use induction to prove a result that contradicts your belief. What you did in your discussion, was to try to prove the very belief that you hold ... and yes, that is not going to help you show you're wrong. That is, you basically do:
Assumption: $\forall n: n=1$
Now let's use induction to show that $\forall n: n=1$
Base: $1=1$. Check
Step: Assume $k=1$. Now, will $k+1=1$? Yes, because of my asumption! So, check!
So, there, we have proven it: $\forall n: n=1$
... well yes ... you can prove it because you assumed it. In fact, any method can prove your result once you assume it. But that does not mean that induction (or any method) can not show you are wrong. As long as you believe other things about numbers, then we can try to use induction to prove other claims .. claims that can end up contradicting your original belief.
Now, given that you believe that all numbers are $1$, I really can't say what other claims you would be willing to agree to ... though probably not too many of the ones that mathematicians hold about numbers, because you effectively end up saying that there is exactly one number: $1$ . That is, you would probably reject the notion that there can be different numbers, and once you rteject that, there is not much left ... But let's assume for the sake of this argument that you do agree to the following claims (after all, you do seem to be ok with induction as a proof technique, and so you do seem to agree that you can reach all numbers, starting with 1, by keep adding 1):
$A$. $1$ is the 'first' natural number. That is, $1$ is not the 'next' number relative to some other number.
$B$. Every number has a 'next' number
(Again, I think that if you believe that all numbers are $1$, and you have some sympathy to the process of induction, you might agree to these as well. That is, presented with these, you might say:
"Yes, $1$ is indeed the first number. It is what starts the induction, so there is no number before it. In fact, it is the only number, since all numbers are equal to it, so what other number could possibly be before it. Ha ha! So sure, I can live with $A$"
"And yes, every number has a 'next' number. This is what we use in the inductive process. In fact, the 'next' number from $1$ is of course just $1$, since all numbers are equal to $1$. Again, no problem there! So sure, I can live with $B$")
OK, but now you will in fact have a contradiction on your hands! By $B$, $1$ has a next number, and by your belief that all numbers are $1$, this will just be $1$ again. But that means that $1$ is the 'next' number from $1$ ... meaning that there is something (namely $1$) that $1$ is the 'next' number of! And that will contradict A.
Of course that is not using induction, but I can use induction if I wanted to. That is, let's use induction that on the basis of your original belief that all numbers are $1$, and that ($B$) all numbers have a 'next' number, it follows that, contrary to A: $1$ isn't just the 'next' number of some number, but of *all numbers! Here is the inductive proof:
Base: $1$ has a next number, and since all numbers are $1$, that next number has to be $1$. So, $1$ is the next number from $1$. Check!
Step: Inductive Hypothesis: Suppose $1$ is the next number from $k$. Now let's show that $1$ is the next number from $k+1$. Well, $k+1$ is a number and, by $B$, has to have a 'next' number. But that next number is of course just $1$, since all numbers are $1$. So, $1$ is the next number from $k+1$. Check!
OK, so now we have proven that $1$ is the next number from all numbers. But wait! That contradicts my belief in A, which says that $1$ is the first number, and thus has nothing 'before' it, because I just proved that everything is 'before' $1$! OK, so since the contradiction was derived from my original belief, $A$, and $B$, I need to reject one of these. Well, I really want to hold on to $A$ and $B$, so there: my original belief that all numbers are $1$ must be mistaken!
OK, so there you have it. I used induction to contradict your original belief, and to indeed possibly make you believe otherwise. Now of course, you could also end up rejecting belief $A$ or $B$ and stubbornly hold on to your original belief, but like I said, there won't be much else you can hold ...