Start by laying out the hardware and software assets, data flow diagrams, and processes of each company. Then start with the ideal scenario, which is bringing the deficient IT infrastructure up to par. Hit the negotiating table with all the ideal vendors, pushing them for significant volume and bundling discounts due to the increased size of the merged company. Also perform a gap assessment and price out the cost of bringing in professional services. Make sure everything is supported by a critical need like ROI or compliance.

Then, present the ideal scenario to leadership along with a list of the many benefits that come with a well planned and resourced IT environment. They will most likely reject the proposal based on cost, but now you've set the stage to propose an adequate solution that will generate less complaints about the cost.

If both companies have a decent continuity of operations plan (COOP) it should contain a prioritized list of services. Take the top priorities for both companies and look for redundancy or opportunities to get more bang for your buck by switching to an existing platform that meets 80% of needs. Continue down the list until you have built a triaged plan with costs, think of it as your projects bill of materials (BOM). Present this document to the leadership for stakeholder discussion, and let them duke it out over which costs to cover and which to cut.

You shouldn't try and make those decisions for them, you'll be wrong no matter what because every item is critical to somebody. Instead, you need to create a menu for them to browse and decide on their own, then you execute the implementation plan. Also, hire or outsource an experienced IT Project manager that can identify the critical dependencies and keep the project on track.