This is problematic. Your employer will need acceptable evidence of your right to work.
The first such document you can get is a certificate of application for a residence card, but to apply for that your wife must be a "qualified person." That basically means that she must be working, looking for work, studying, or self sufficient. There are a few other requirements as well.
Unless you have enough liquid assets for her to qualify as "self sufficient," the easiest thing will probably be for her to get a part-time job or set herself up as self employed. Other options, including the "self sufficient" option, require you to buy "comprehensive sickness insurance cover."
You could also seek authorization under the Immigration Rules, independent from your wife's EU citizenship, in the form of an appropriate work visa.
Update: I've just noticed a recent blog post that points out that the employer checks are not actually mandatory, although many think that they are. In fact the checks serve as a defense if one is charged with employing someone who is not authorized to work, but the offense is the employment of an unauthorized person, not the failure to check ("an employer who employs a person who does not have permission to work but who has carried out checks in accordance with Home Office guidance is able to benefit from a statutory excuse against a civil penalty").
If the employee is authorized to work, there is no offense. If you can find an employer who will employ you without a residence card or certificate of application, you can work legally, but after your wife has been in the UK for three months, she must be a "qualified person" for you to have the right to work.