Can you provide more information?
Location (You, client and designer (City, State)), programming language and frameworks.
Is this simple or advanced work in your opinion?
You will need to add money to cover the following and should have different rates for your designer depending on the level of effort, experience they have and the skill set required.
You need to get paid for: Taxes, food, rent/mortgage, gas, vacation, medical, dental, sick days, vision, savings, rainy day money, retirement, investments, entertainment, software, hardware, an accountant, other businesses expenses and any other things you may want or need for your job. So these should be added to that final rate that you charge the client per individual.
Since they want to take you on full time I recommend forming a business (at least an L.L.C.) and have your designer and anyone else you need as a contractor or employee to you. This would allow you to hire more people and get a higher rate than what you are receiving now along with more flexibility in growing with the client.
I recommend creating a chart with all of your current and possible future expenses for an entire year so you can get a better look at where the money will disappear too. As there might be some things that are now showing up for you now that you might only see when you have everything in front of you.
For your contract I recommend preparing things to take to a business lawyer first by breaking it down into separate projects and pay rates for you vs the designer. This can be the same rate if you want it to be but normally someone doing development/engineering/programming/architecture work is paid a higher amount than someone doing design unless it is high end design(architectural drawings, advanced 3D simulations, etc.).
Work up a summary of work along with official documentation on what is expected for each role, the minimum hours per week, business hours, what you require from the client and milestone deadlines, what the penalty costs are if deadlines are not reached or required information is not given, along with wording on an agreement for changes that occur and what the cost of said changes are after x amount of time. Do you supply your own hardware, or will the client supply it for you. As depending on what is required you may be classified by law as an employee of the company or a contractor, which you will need to check with an attorney about.
Once you have a draft ready have an attorney take a look over what you have so you can make sure you are covered legally with the business your are working with along with any of your contractors. Also take a look at the rates of people along with their benefits on popular salary sites so you can get a better idea of what your rate and your designers rate should be set at.