fix: create the routing cache per router instance so apps can be garbage collected

[uttam12331]

Description

ASGIRouter.handle_routing was decorated with a class-level @lru_cache(1024). A class-level cache is a garbage-collection root: every router instance — and, through router.app, every Litestar app with all of its state — stayed strongly referenced by the cache forever. Long-running processes that create many app instances (most notably test suites) accumulated every app ever created, as reported with a gc-based MCVE in the issue.

This PR binds the cache per instance in __init__:

self.handle_routing = lru_cache(1024)(self._handle_routing)

The cache then only participates in a self-referential cycle (router → cache → bound method → router) that the garbage collector reclaims once the app is no longer externally referenced. Details:

• The call surface is unchanged: self.handle_routing(path=..., method=...) works as before, and cache_clear() / cache_info() remain available on the per-instance bound cache (the workaround from the issue keeps working).
• handle_routing is added to __slots__; the uncached implementation lives on as _handle_routing.
• Per-call overhead is the same one lru_cache wrapper as before.

Verified with the issue's MCVE: on main, one Litestar instance survives del app; gc.collect(); with this change, zero survive. Added test_routing_cache_does_not_prevent_garbage_collection, which fails on main and passes here; the full tests/unit/test_asgi/test_asgi_router.py suite passes (14 passed).

Disclosure per the contribution guidelines: this fix was developed with AI assistance (Claude Code), with the leak and fix verified locally as described above.

Closes

Closes #4876

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📚 Documentation preview 📚: https://litestar-org.github.io/litestar-docs-preview/4930

[codecov]

Codecov Report

❌ Patch coverage is 50.00000% with 1 line in your changes missing coverage. Please review.
⚠️ Please upload report for BASE (main@6d07038). Learn more about missing BASE report.

Files with missing lines	Patch %	Lines
litestar/_asgi/asgi_router.py	50.00%	1 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #4930   +/-   ##
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  Coverage        ?   67.30%           
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  Files           ?      293           
  Lines           ?    15230           
  Branches        ?     1728           
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  Hits            ?    10251           
  Misses          ?     4832           
  Partials        ?      147           

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