Since you refer to this as a "high profile interview," it is not unlikely that the interview is going to be re-aired or rebroadcast by others. If in fact you own the copyright, therefore, it would be in your interests to register it with the United States Copyright Office whether or not you proceed against the channel or channels that already aired the interview without your permission.
I make this suggestion because, presuming you are entitled to register, it would enable you to seek "statutory damages" and attorneys fees in the event of a FUTURE infringement - i.e., one that commenced after the effective date of the registration. Statutory damages of up to $ 30,000 can be awarded for a single infringement; up to $ 150,000 in the event of a willful infringement. Actual damages, on the other hand - which are awarded in cases in which the infringement commenced prior to registration - are generally much more modest. Sometimes, they are nominal.
Rather than proceed on your own, however, I agree with my colleagues that you need to retain copyright counsel to review the facts and circumstances. Counsel can help you effect the registration and decide whether a claim or demand letter is warranted.
Disclaimer: This answer does not itself establish an attorney-client relationship. It is for general informational purposes only.
There aren't enough facts here to know what if anything you may own rights to and how it was used and by whom. I assume the subject of the interview made this a newsworthy event which would be an exception to the state publicity rights laws, so the question is about your copyright to the footage and the network's use. There's good precedent for a claim of copyright infringement when even neworthy footage is used by news outlets, such as the Zapruder tape of the Kennedy assassination, but there are many questions here.
Did you work for this network, were you hired to produce this footage and/or did they pay for this footage? If so they may own its copyright, not you. How did they get your footage, if you didn't give them a license to use it? Did you have an agreement with them that they'd compensate you for using it? If parts of the interview had already been aired on other networks and displayed in print outlets, did you have deals in place with those other networks and outlets for use of your footage?
Bottom line is that you need to hire a lawyer to discuss the details with and to negotiate a deal for you, if approptriate.
Disclaimer: Please note that this answer does not constitute legal advice, and should not 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. This answer does not create an attorney-client relationship.
You may have a viable claim for copyright infringement, although your failure to register probably will preclude you from obtaining statutory damages. In order to bring a copyright suit, you would need to register copyrights in the footage. Further, you will need to prove actual damages. There are various ways to show actual damage---but there must be solid proof of damages. For example, you could try to show that you lost profits that you could have made by selling your interview to someone else---the footage that the infringer used caused you to lose opportunities to sell the interview to another channel. In order to prove this, you will have to have concrete evidence as to what stations would have bought the footage and for what price---you cannot just speculate. Alternatively, if you cannot show lost profits, you can try to prove the amount of the royalty (or fee) that you would ordinarily have been paid by a television network that used the interview--once again you cannot prove this claim by speculation--you will need to show what others paid you for the footage and/or what others have paid in the past for similar footage. You need hard evidence of this in order to obtain damages. You also could try to force the network to disgorge profits that it made as a result of using the footing, but this can be very difficult to prove---you will have to show that but for use of the footage, the network would not have made as much money on the show. However, if the network could have made the same point, or covered the same or similar ground without your footage and/or if the footage was only a small piece of a larger broadcast, it will be difficult if not impossible to show that then network made large amounts of money because it used your interview.
Further, the mere fact that the network apologized for using the footing does not necessarily mean that it has no defenses against a copyright claim. It would probably try to argue that you gave it permission orally to use the interview and/or that one of your agents did. Alternatively, it would argue that the use of the interview was for journalistic, educational or artistic purposes constituting fair use and protected by the first amendment.
Also, I am not entirely clear from your question that you actually own the footage. You indicated that you have the original tape and invoices of the shoot. However. ordinarily the photographer owns the copyright in footage such as this. Absent a license agreement, work for hire agreement, and/or assignment from the photographer to you, you may not actually own the footage. Obviously, you would need to establish ownership of the footage by showing the existence of such an agreement. In the absent of such an agreement, the mere fact that you paid for the shooting of the interview does not mean you own it.
The bottom line is this---you may have a viable claim but there are many hurdles which you must overcome. Even if you do have a viable claim, it may not be a pot of gold for you. And the only way for a lawyer to give you definitive and meaningful advice would be to sit down with you, conduct a detailed investigation, and then help you determine whether it is worthwhiile to pursue this claim. You certainly cannot expect to figure this out on Avvo---these are complex issues that need to be carefully analyzed and reviewed with an attorney. Thus, your next step if you want to pursue this is to retain copyright/IP counsel.
All fluff aside, you want money from the company that showed "your" videotape on television.
Ok, but we all know the truism that in order to make money you have to spend money. So, hire a copyright litigator to assess the situation and, if legally supportable, he or she will demand money from the company.
Is your claim against the company legally supportable under copyright, or other relevant, law? Will it cost you more to get money from the company than the dollar amount of that money? Dunno. No one but your own copyright attorney can answer these questions. Will that attorney work for free? No. Will he or she take your case "on contingency?" Very, very doubtful. But you can try. Good luck.