Cost of Documentation

I am writing a proposal for my company to get some outside help documenting some software I have written. Currently, there are over 20,000 lines of code in a web application. Please give me some rough numbers on what to expect it to cost if we were to hire a company to write the documentation for us.

Thank you,
Jake

The number of lines of codes in an application has no bearing on the cost of documentation. It just doesn't relate.
First of all, you need to identify what documentation you need. If you have already produced the software, you have skipped over the most important documentation: Requirements and Architecture. This is a common mistake project managers make and it leads to a poorly designed product and does not serve the user's/customer's needs. Documents created during the design and development phases are the guiding blueprint for the project. They provide a framework and a standard the entire development team must adhere to. When design documents are not created, the user (the prime focus of the requirements docs)is not served. Engineers and programmers are so enamored of features and functions, they often forget that the user does not have the same level of techno understanding they do, and they turn out a product that is troublesome and difficult to use. There is no accountability for design and implementation. To produce any good product, you must have a plan and you must stick to it. Tech writers are the keepers of the plan.

I have been brought in on many projects to write documentation for software near completion and it is a frustrating and often unproductive task: Subject Matter Experts (SME)s have often moved on to other projects and are not avaialable for consultation. It's like closing the barn door after the horses have already escaped.
If I use a copy of the application to write a user manual, I may find flaws in the functioning, things that could have been caught earlier; with some design accountability and thorough test scripts.
My recommendation is to consult with a tech writer: define exactly what you want them to produce and work out a timeline for getting it done. From that you can calculate the approximate number of hours the tech writer might need to accomplish it. Only then can you know what the cost is going to be.