refactor(api): migrate from sdk to ddp-client

[this-is-varunxyz]

#Brief Title

refactor(api): migrate from sdk to ddp-client
Acceptance Criteria fulfillment

[x] Removed deprecated @rocket.chat/sdk dependency

[x] Added @rocket.chat/ddp-client and implemented EmbeddedChatApi connection logic

[x] Refactored real-time subscriptions (room messages, typing status, user activity) to use DDP

[x] Added dependency resolution for @rocket.chat/core-typings to fix build errors

Video/Screenshots

N/A - Internal API Refactor (No UI changes)
#PR Test Details

Description: This PR migrates the core API logic from the deprecated SDK to the supported ddp-client. This ensures better long-term support and fixes dependency resolution issues.

How to Test:

Switch to this branch.

Run yarn install (Verified that resolutions in root package.json fixes the core-typings error).

Navigate to packages/api and run yarn build.

Verify that the build completes successfully and dist/index.js is generated.

Note: The PR will be ready for live testing at https://rocketchat.github.io/EmbeddedChat/pulls/pr-<pr_number> after approval. Contributors are requested to replace <pr_number> with the actual PR number.

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