If the lawyer you want to sue has committed malpractice, that means that "but for" their acts or omissions, you would have gotten a different result. It requirs proving the trial within the trial, so the underlying case has to be something you would have won.
Lawyers will sue other lawyers fro malpractice, but usually on an hourly fee basis and not on a contingency basis, with the lawyer being paid out of your eventual recovery. Contingency deals are usually reserved for personal injury cases with clear liability and substabtial damages, and fopr copyright and patent infringement cases, with clear liability and substantial
damages, and sometimes for employment cases, with strong liability and damages and statutory fees.
Maybe your case is an exception, and the lawyer you want to sue has a big malrpactice insurance policy, and maybe your case is very strong because the lawyer clearly screwed up, ike missibg a filing deadline, and maybe you have big damages.
Don't expect to find anyone to work for free--- we all have bills to pay and can't afford to work pro bono for everyone who asks (and there are lots of requests). If you're eligible, maybe you can get a free lawyer from your local Legal Aid office - those lawyers do get salaries from the government.
PLEASE READ THIS BEFORE YOU COMMENT, EMAIL ME OR PHONE ME. I'm only licensed in CA. This answer doesn't make me your lawyer, and neither do follow-up comments and/or emails and/or phone calls, and you shouldn't expect me to respond to your further questions if you haven't hired me. We need an actual agreement confirmed in writing before any attorney-client relationship is formed. This answer doesn't constitute legal advice, and shouldn't be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.
You asked for an attorney who will sue another lawyer probono. Pro bono means "for free" I believe what you are looking for is an attorney who will take the case on "contingency". This means the lawyer who takes you case will get a percentage of your settlement, usually 25 - 45%. I doubt any attorney would be willing to take your case pro bono when you stand to collect a monetary judgment in the case.
Try reposting your questions asking for an attorney who will handle a malpractice case on contingency.
I am an attorney licensed to practice in Texas, Louisiana and before the United States Patent and Trademark Office. Laws very from jurisdiction to jurisdiction and the above may not be an accurate assessment of the laws for your specific jurisdiction. The above should be taken as general guidance and not specific legal advice. For specific legal advice you should seek a licensed attorney in your jurisdiction practicing in the area specific to your issue. The above does not constitute or establish an attorney client relationship. If you wish to receive specific advice about your legal issue, then contact my office to schedule a consultation.