Javascript is broken with FastAPI sub-application.

**Describe the bug**

When you [mount the fastapi app as sub-application](https://fastapi.tiangolo.com/advanced/sub-applications/#mount-the-sub-application), the js in `/dev-ui/` stops working.

**To Reproduce**

1. Make the agent
```python
from pathlib import Path
import uvicorn
from fastapi import FastAPI
from google.adk.cli.fast_api import get_fast_api_app
app = FastAPI()
adk_app = get_fast_api_app(
agents_dir=str((Path(__file__) / '..').absolute()),
session_service_uri="sqlite:///:memory:",
allow_origins=["*"],
web=True,
)
app.mount("/adk", adk_app)
uvicorn.run(app, host="0.0.0.0", port=8081)
```
2. Run the code
4. Visit http://localhost:8081/adk/dev-ui/
5. See the error Http failure response for http://localhost:8081/list-apps?relative_path=./: 404 Not Found

**Expected behavior**

There would be no error, proper subpath would be used.

**Screenshots**

**Desktop (please complete the following information):**
- OS: Windows 11
- Python version(python -V): Python 3.12.8
- ADK version(pip show google-adk): 1.11.0

**Additional context**
Add any other context about the problem here.