1) The President is either the tool of Congress or simply the head of the Cabinet. The Executive branch becomes about implementing laws and directives of Congress, not about being the Commander-in-Chief, Prosecutor-in-Chief, etc. His most influential role might be in brokering compromises between the Senate and House, however.
2) Cabinet officers rather than being picked largely by the President are picked by Congressional leaders, particularly Senate leaders. Presidential nomination is largely a formality.
3) The State of the Union remains a written report submitted by the President to Congress, rather than a speech.
4) There is a showdown between the House of Representatives and the Senate. Their powers, if presumed to be unchanged, are relatively co-equal: the Senate has pre-emeninet powers over foreign relations by virtue of the advice and consent clause as well as over the Cabinet and thus the Administration. The House has a wide birth over domestic affiars, due to the requirement that all appropriation bills originate with them. If the Senate remains indirectly elected, I would expect the House to be particularly vociferous in demanding that it in fact must originate all legislation since all legislation at some point directs the expenditure of money and hence can be seen as an appropriation. (This is similar to what happened OTL to the interstate commerce clause.) If so, then the House become the predominant branch. When direct election comes around, it may be election by the House, rather than by the Senate. This would bring the old Virginia Plan of Madison directly into operation.
5) I would probably expect to see a larger House of Representatives than we have OTL. With no election of the President, people will want more responsiveness out of the House than its traditionally small numbers allow. The leadership of Congress will agree because larger numbers of Congressmen decrease the marginal ability of any single one or any small group to stall legislation and thus buck party unity. A larger House allows the Speaker to more effectively run the legislative agenda.
6) There would strange effects as well on the terms of Congress. Jansen's original suggestion is that the President is appointed to a four year term and can only be impeached for gross misbehavior, as per OTL's Constitution. If in a mid-term election, the House changes hands what does the new Party do with the President? I think this question is largely muted, because I think the President will be largely apolitical, like the President of Israel, Germany, or Ireland (and perhaps Poland, though there's a bit more power there).
7) In short, something like a Parliamentary system evolves within the confines of the American system. The major differences are: regular elections, the separation of executive offices and the legislature (rather than Cabinet memebers being part of the legislature). I think this would have the effect, however, of decreasing the importance of the Federal government and the extent it becomes a national government. Congress has never had much respect from the national at large because it becomes a last hive of folks who are all bent on the benefit of their own districts and constantly looking to raise money for their next campaign. The Speaker of the House and the leader of Senate (which might be organized differently in such a system) would not fulfill the same kind of role as national figure. The TTL President would be unifying, as a broker of compromise, but he would be largely ceremonial. Without this feeling, there will be in, IMO, greater reluctance to expand the scope of the federal government. I should also add that parties work in very different manner than OTL American parties: there may be more of them, for starters, particularly regional ones. There will also be more of divide between elected officials (politicians) and appointed officials (adminstrators). The Cabinet offices that we think of as political offices will be less overtly so as will the Presidency.