the meaning of the last two lines quoted;

>Feels [sic] his Deserts with honest Glow,
And all is Self‐improvement know.

..depends upon the preceeding two lines;

"Then like his father too, he must,
To his own former struggles just,"

I think Jane wants to point out the just deserts (to receive what one deserves/a reward for what has been done) received honestly by the father in good spirit as a way of self improvement should be emulatd by the son as well in the full knowledge of its benefits.

I could be completely off the mark here. :)

>"Bet, my b not come to bide."

This one is difficult. I feel it has to do with the curly hair showing just above the window sill during the act of peeping into the nursery and saying "Bet, my b not come to bide."