This is actually a very complex question because you have to read between the lines to find the answer. After examining the text, I believe there are actually three reasons that Annie gives to the Kellers in order to convince them not to put Helen in an asylum, and two of them have to do with both Annie and Helen having strong characters. In short, Annie is determined, the asylum contains unspeakable horrors, and Helen is strong. Now let's look at each in turn.
This conversation begins with a misunderstanding when Annie, in frustration, exclaims that "It's hopeless here." Annie means that it's hopeless in the Keller household because of the pity the family has for Helen. The Kellers mistakenly think Annie is giving up on Helen and, in their pleading, mention the asylum they considered before Annie arrived. That is what begins the conversation where we can find these three reasons.
First, Annie finally corrects the Kellers by telling them she is determined to help Helen.
Kate: You said it was hopeless.
Annie: Here. Give up, why, I only today saw what has to be done, to begin!
Now that is a determined woman! Annie, then, out of her own determination won't allow the Kellers to put Helen away!
Second, Annie uses the actual horrors of the state almshouse as her second reason. She mentions everything: rats, unsanitary conditions, disease, prostitution, rape, severe mental illness, death, and children playing among the dead bodies.
The room Jimmie and I played in was the deadhouse, where they kept the bodies till they could dig the graves!
In reality, if you wanted to count each of these reasons separately you would have a plethora of reasons why not to send Helen to an asylum!
Finally, Annie mentions Helen's own strength. This particular reason is stated point-blank: "I don't think you need send Helen there. She's strong enough." Annie, then, has confidence enough in Helen's strength of character to know that Helen could, in fact, learn to conquer her condition.
Thus, as you can see, there are multiple reasons that Annie gives in opposition to Helen being sent to an asylum. Thank God that Annie was right, ... she was strong enough to convince the Kellers!