Annie gives two reasons in The Miracle Worker why Helen should not be put into an asylum. What are they?

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.

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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.

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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!