How can I enforce my visitation rights as a father when I live in the UK and my daughter and ex wife live in Illinois just outside chicago?

There are a number of things that can occur.  The first question to address is where it is that you were divorced?  Also, did you live in Illinois or had you already relocated to the UK?  This is important as the divorce decree is the place to start for your basic rights for visitation (as the decree should have specific language detailing visitation).  The question then turns to one of enforcement (if the relocation to UK is referenced in the Judgment) or modification (if you need to address this change in circumstances).  Either will require a return to court in Illinois (as this presumes this is where the divorce occurred).  The next question is one of timing.  It is highly unlikely, but possible, you could hire an attorney and secure a hearing prior to Christmas.  There is a very short window, even for an emergency motion.  It is, however, possible and you could also try to secure such a motion yourself.  Securing counsel immediately would appear to be essential.  Even if only for a consultation. You may find that you need to hire an attorney, file pleadings and then appear at either mediation or court or, possibly both, to secure visitation.  Each county has its own rules involving visitation disputes, which is another reason to secure the advise of an attorney experienced in such matters.  As to the kidnapping issue raised by your ex-wife, provisions, such as stipulating to jurisdiction in the Hague regarding any return/kidnapping issues, may be an option and should also be discussed with an attorney. Also, any return to court could bring with it some unintended consequences, such as petitions for Rule against you for asserted non-compliance with the decree and/or petitions to increase support based upon asserted increases in your income.  To insure that you have fully assessed the situation, all of this should be discussed in a consultation with an attorney.