Biomedical engineers design and develop biomedical devices and systems, from pacemakers to MRI machines, prosthetics to to advanced wound care. It draws heavily on biology and medicine, but is quite interdisciplinary and also draws on other fields, depending on the type of thing you work on (physics and electronics for MRI, mechanical engineering for prosthetics, etc.). Biomedical engineers work for companies of all sizes, from startups to Medtronic.
Chemical engineering can be something of a "gotcha" field, because how it is named is not consistent with other engineering fields. Most people assume that chemical engineers design and develop chemicals, the way electrical engineers do electronics and structural engineers do structures (but it's actually chemists who do that). Chemical engineers actually work on mass-production systems and scale-up. They design, develop, and operate large plants and highly integrated systems. It is the most interdisciplinary engineering field of all (save perhaps industrial engineering, which I don't think is really engineering at all: it's management!): the first two years of the ChE degree program are like a liberal arts in the sciences and engineering, but then in the latter half you learn how to size pumps, cooling towers, and reactors. It is very mathematical. The largest single sector in which chemical engineers work is the petroleum industry: refining petroleum, cracking hydrocarbons, and making polymers and pharmaceuticals, but some of my friends work in almost every industry that mass-produces its products, from computer chips and cars to soda pop and potato chips. One of my friends makes Depends undergarments, another makes Velveeta. In most cases, a chemical engineer sees a production line through from start to finish: they design the production plant, oversee the acquisition, installation, assembly, and integration of the equipment, and then run the line (oversee its operation and maintenance) for as long as it remains relevant. ChE pays extremely well and it is easy to get a job in, but you are on call at all hours once the line launches because you are the expert on the big picture, and get called when it isn't working right and nobody else can figure out why in the wee hours of the night. (Some ChE specialize in just design, build, or operate, though.) Patents in ChE relate to making the production lines more efficient, in terms of speed, reliability, and economics. The vast majority of ChEs work for larger companies.
Hope that helps! Kaitlyn's advice about looking at job postings is solid, too: but look for common threads in the postings, not the exceptions!