Is the planning phase the same for every client? Does the size or the complexity of the project have more to do with the time required to plan the project?
Way, way less than you or the client seems to think!
See https://en.wikipedia.org/wiki/Big_Design_Up_Front -- having a distinct planning phase that precedes design, development, etc. is a bad idea for a variety of reasons.
Instead, I recommend a more iterative approach (doesn't need to be a formal methodology like SCRUM) where all parties understand that having a product under development is when a lot of the discovery happens, not when it's all mock-ups.
In both client work and in-house development, it's practically a given that project requirements change over time (e.g. "Ah, well now that I look at the website and am actually clicking around on it, I would rather ...").
So, instead of getting upset about this, or trying to fight against its inevitability (we're human, no way we can perfectly forecast something as complex as a software project) I heartily recommend *embracing* the fact that requirements change frequently, by adopting an approach that eschews a "planning phase" for frequent revisions by your client.