This is a perfect example of why you use remodel cans. I'd patch that, then locate all the ceiling joists in the area. Drill for your cans avoiding the joists, fish the wire to the cans, hook 'em up and you're done.

Ask around for referrals for electricians who are willing to do small jobs. If you have a flexible schedule, tell the electrician he can put you as a fill in job. Also tell him if you want this done soon or some longer time period. Have the area cleared out and a defined project, like 3 lights in this space and 2 over here, but not so rigid in the details that you end up actually costing more. Being an easy customer to work for will increase your chances of getting a good job at a reasonable price. If you have a very rigid schedule with limited availability, expect to pay for the privilege on having the project done at your convenience, not his. I would be very cautious of any young guy who is willing to give you a fixed price as he might not have had enough jobs with hidden surprises. If you find someone you are comfortable with, you should be able to get a reasonable expectation of the cost in advance. Not a fixed fee but more like the fixtures will cost this much, other materials will be about this much and it will probably take me a couple of hours so you are looking at somewhere around $X, unless I find some safety issues that I need to fix. That is as close to a fixed price you will likely get unless the bid is from a cocky youngster or expensive enough to still make money despite the obstacles that may exist. If you find the right person to do the work, the time and material billing won't be a problem.

Yes - recessed lights should be insulated to prevent this exact thing. Most homeowners do not realize the amount of heat these types of light put off and the damage that heat can do to their home & HVAC systems.

Patricia, LED canless recessed lights perform the same function as canned lights, but because they are LED and don't generate heat, insulation isn't an issue, they are thin. They are not rope lights and are not vanity lights, they are used for general lighting same as canned lights but are much thinner, don't have to be attached in attic, the LED and trim come with a small junction box, can be used new construction or remodels.