Firstly, Tom, I applaud your decision to bring your child up to speak more than one language. Like my brothers, I was born and raised in Britain to a Danish mother and a Polish father, and at home my mother spoke only Danish to us until I was about ten years old (unfortunately my father was not motivated equally strongly to teach us Polish).
The result is that not only were we able to communicate effectively with our Danish relatives, but our previous experience of a language other than English made learning new languages easy when they were introduced into the classroom at age 11. I am in absolutely no doubt that a child brought up bilingually has a significant advantage in this respect, and is probably less liable to be culturally conservative (in other words, may be more willing to try new foods, socialise etc).
I suspect you may be correct to believe that you cannot successfully teach a baby or a child a foreign language unless you speak it to them exclusively. I can point to some evidence for this belief in the shape of one of my nieces (soon five years old), who is being brought up to speak Danish by my brother, who is the only Danish speaker in her home.
That's the theory, anyway. In practice, my niece understands Danish but will not speak it, despite the fact that my mother too (the child's grandmother) will often try to encourage her when they are together. My brother's command of Danish is not equally strong in all areas, and possibly partly because of this, he has not been communicating with her solely in Danish, especially since she started going to kindergarten. It does not help that there are no Danish-speaking families in his social circle. This means that my niece only ever hears the language spoken by a couple of close relatives. In that regard it is a different situation than when a foreign family settles in a community where there are many other families that share the same cultural background and are able to mutually reinforce both the cultural and linguistic aspects of their identity.
For this reason, in your case it might be advisable to expose your daughter to other people speaking English besides yourself (native speakers if possible). This will help everyone to feel that the situation is less artificial, especially since you too are a non-native speaker of English. Plus, of course, it will help to smooth off any remaining rough edges in your own spoken command of the language.
Concerning your original point: English makes considerably less use of diminutives and pet names than does Polish, which is probably why there have been relatively few suggestions made here thus far. Many of those which have been made are more likely to be said about
a small child than to
it. (For instance, in England "little tyke" is a derogatory term which it would be inappropriate to address directly to a child unless the intention was to brutalise it.)
Reply from Erik Kowal ( - England)