How can I report the income received from the partial sale of my business to bring on a partner?

assuming you started with a sole proprietorship (no S-corp election) you have now formed a partnership and there is now a need to file form 1065 - partnership return.   if the money from the other partner was left in the business you do not have a taxable event.   if you took some or all of the money out it's capital gain to the extent it exceeds your basis. I strongly recommend that you use a tax pro to make sure there is proper reporting. 

the profit before the partnership was created would still need to be reported on your Schedule C. the partnership income and expenses are for the period starting on the date the partnership was formed.  

In a nutshell, assuming you had a single member LLC before the sale, you now have a partnership after the sale. Therefore, the single member LLC is closed/dissolved, and a new multi-member LLC is formed. Dealing with this "CORRECTLY" is important. Especially if your state taxes personal income. I highly advise you seek the services of a tax professional for this year. Mistakes can be (and will be) costly later down the road. The cost of mistakes will make the cost of professional help seem like a pittance in comparison. So please, seek professional help for this year. THe cost of professional help may also qualify as a business deduction in the newly formed partnership/multi-member LLC also.