Include results from `--extendedDiagnostics` when using the compiler API

Nobody has claimed this yet.

### 🔍 Search Terms

extendedDiagnostics, compiler API

### ✅ Viability Checklist

- [x] This wouldn't be a breaking change in existing TypeScript/JavaScript code
- [x] This wouldn't change the runtime behavior of existing JavaScript code
- [x] This could be implemented without emitting different JS based on the types of the expressions
- [x] This isn't a runtime feature (e.g. library functionality, non-ECMAScript syntax with JavaScript output, new syntax sugar for JS, etc.)
- [x] This isn't a request to add a new utility type: https://github.com/microsoft/TypeScript/wiki/No-New-Utility-Types
- [x] This feature would agree with the rest of our Design Goals: https://github.com/Microsoft/TypeScript/wiki/TypeScript-Design-Goals

### ⭐ Suggestion

For users of the TypeScript compiler API, it would be extremely useful to include the performance statistics obtained when using the `--extendedDiagnostics` compiler option.

From the review I've done in this codebase, it seems like this performance information is only reported when invoking `tsc` directly.

### 📃 Motivating Example

"The compiler API can now report performance information when enabling the `extendedDiagnostics` compiler option"

### 💻 Use Cases

1. What do you want to use this for? Investigating performance variations in a project that interacts with TypeScript using the compiler API.

Start by tracing how the `--extendedDiagnostics` option reports performance information when invoking `tsc`, then compare that path with the compiler API. Define how the compiler API should expose those statistics and verify that a compiler-API consumer can retrieve them for performance investigation.

Written by the indexing model from the issue text.