How do you make a site responsive when you don't have the luxury of starting from scratch?
It's not about you
The most important thing is your site - you need to keep it running within the context of the business and your users. Keep the priorities in check.
Make a plan
It will take time. Be persistent and patient. Understand other tasks that are going on concurrently. Sometimes you won't work on it for a while - make sure this is part of your plan.
Define the constraints, you don't have the luxury of fixing everything. What's the scope? You might not be changing content or visual direction.
Do one thing at a time, make sure that tasks are done in the right order - think about dependencies. By working through a list of tasks makes it feel like you are achieving things.
Get as many devices as you can - look at your analytics. Should only be for testing and always available. If you're getting bug reports from a specific device/OS then get hold of that.
Evolution not revolution. Re-use things. Identify the good things that you can carry over. Be clever about the time you have available. Use things that others have already done. Think about what you could do in an hour.
Standardise across sites. Flexibility and standardisation are key. Create a style guide. Having an overview of patterns helps focus on reusability.
Experiment on smaller projects. Use things you have learned there on larger projects. Keep a record of the decisions you make, this allows people to understand things later.
Just do it
Tidy up before you start. Catalogue what's there to look for patterns that already exist.
Set the rules. Define common patterns, be opinionated to give people things to follow in unforseen situations.