I am sorry to have taken so long to respond. But a lot of things have happened and I have been very unhappy (big understatement).
Firstly, can I thank those of you who were suggesting I should take another DNA test. It has clarified things.
After a great deal of discussions, my wife finally agreed to a DNA test. But explained quite strongly that she opposed it now. I didn't take it any further and ask her why. I genuinely don't understand and suspect I never will.
Anyway. Having been given the go-ahead I then had to arrange for the test. This was rather complicated. For a start you can only have a DNA test if you have the written permission of the main carer for the child (it's a legal requirement I think). And as both the child and mother were abroad that made life rather complicated. (could I prove that the letter was written by the mother etc).
In the end I decided it was easier if I chose a testing centre in Poland.
I searched online and read various forum comments and eventually chose a major provider of DNA tests in Poland. Not the cheapest but they have a good reputation.
I flew to Poland. And then all three of us went to the testing centre. Where, firstly, we had to prove our identities. Then all three of us had samples taken. The result is more accurate if all three of us are tested.
And the result was.............. I AM the father!
And the other result is that my life has been completely turned upside down.
When we got the results the Polish mother looked at me with a hard face as if to say 'why would you ever be so awful to think that I would lie about such a thing'. She is very angry. And still is.
My wife was devastated. I cannot put into words how upset she is. It really is heart wrenching to see her like this. I feel even more guilty and miserable than ever.
My wife still point blank refuses to let me tell our English daughter about her Polish half sister. In fact she is more resolute than ever about this.
This has resulted, bizarrely, in the Polish mother refusing all maintenance payments from me. She is very angry about the whole thing. She thinks it is ridiculous that I must only communicate with her via my wife and that my wife is controlling the maintenance payments.
So, in conclusion, she is my daughter. But my wife and the Polish mother are both very upset and angry with me. And my English daughter has not been told and probably will never be told.
Thank you again to all of you who have been so kind and helpful with advice and being so understanding.
I have no idea what the future holds.