This phase I/II research study is evaluating a combination of drugs called paclitaxel and ruxolitinib as a possible treatment for inflammatory breast cancer. Ruxolitinib is a newly discovered drug that has been shown to block a pathway (called the IL6/JAK/Stat pathway) that may be important in cancer, including breast cancer. Blocking this pathway may stop cancer cells from growing. Ruxolitinib has been approved by the FDA for patients with bone marrow disease, and this is the first study using this drug in combination with paclitaxel for breast cancer. Paclitaxel (also called Taxol) is an FDA drug approved for breast cancer patients. Paclitaxel works by blocking the small microtubules inside cancer cells and preventing cell growth. Information from laboratory experiments suggests that ruxolitinib might also have effects on breast cancer.These studies have shown that ruxolitinib may make paclitaxel more effective.