Are you trying to remove the glue from the glass?
What part of the table is broken?
Ther are two part anarobic epoxies that are made for securing automotive mirrors to glass that dry clear. These are available at any automotive parts house.
The Liquid Nails will only make a big mess of things. You are correct also it is not strong enough.
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Yes more than enough. Consider that the glue has top hold up to many years of yanking on it, bumping into it, things hung from it, cold, heat, etc. The table legs also tend to have a bigger foot print than a mirror post.