Obviously a lot has to change at the club:
1) They need to find the correct manager who will run the team in a somewhat usual fashion and find the best XI. One of the challenges this season is that the squad rotation across the board was inconsistent and it rarely seemed like players had a chance to spend a lot of time playing with the same players, particularly in the midfield and CB positions, which led to a lot of miscommunications and goals conceded.
2) It was inappropriate but Kane wasn't wrong when he talked about some players not being tough enough and not playing up to their potential. The squad needs to be shaken up and its up to Levy and the new manager to find out which players need to go and who needs to be brought back. There are a few scapegoats like Winks and Sissoko who probably need to go, but the problems extend across the whole club.
3) The team scored a lot of goals this season, second in the league from open play. This was done despite the team finishing 9th in xG. That pretty much tells you the value of Kane and Son, who consistently converted on chances. This was done with an unstable midfield and an often regressive approach to attacking play. Will Kane (and Son) be on this team next year? It looks like that isn't the case, but if they do somehow return, this team should score a lot of goals with a more progressive manager and more stability in the XI.
4) Goalkeeper needs to be figured out if Lloris is really gone. Lloris made a ton of big stops this season and was one of the most valuable players on the team, and there isn't an heir apparent on the roster right now. It is likely whoever is brought in will be a downgrade, which puts even more pressure on the CBs and midfielders to perform at a higher level.
5) Speaking of CB, the combo of Dier/Alderwireled/Sanchez was pretty rough throughout the year. Dier and Sanchez were really mistake-prone, with Dier lacking pace and Sanchez consistently losing forwards on balls that got sent into the box. Part of the problem was again an inconsistent midfield and squad rotation, which produced an incredible number of CB pairings, with Rodan and Tanganga also getting involved at different times. We are a long way removed from Toby and SuperJan just starting every game.
6) The fullbacks were another issue, again with Aurier/Doherty/Reguilon/Davies/Tanganga just rotating around with little consistency. A big part of the problem imo was that Aurier/Doherty/Reguilon are all attack-minded fullbacks and Jose didn't use them like that, which exposed their weaknesses and hid their benefits. Reguilon was generally a pretty strong performer, but had some really rough games that exposed his weaknesses (like his game against Villa). There needs to be some changes with that bunch and also the new manager will have to figure out what combo/style works best with the remaining personnel.
7) Center Midfield is going to see some turnover, with Sissoko likely gone and Winks perhaps right on his heels. GLC and Dele both struggled with fitness and form at different times, although most people still seem to be bullish that GLC is still the man for the job. Ndombele had his moments under Jose but also completely disappeared once Mason took over and you wonder if it is a fitness issue or an attitude issue because it made little sense to keep playing Winks over him. He remains an enigma for the time being. Hojbjerg was one of the best players this season and played every minute of the EPL season, which eventually wore him down, but I think most people agree he's a reliable starter for this squad moving forward. Oliver Skipp had a strong season on loan at Norwich and will be one to watch next season as he could crack the rotation, which would be great because this team needs young players with exciting futures.
8) Wingers were a position of strength this season, with Son having his best season as a professional. Lucas/Lamela provided different things off the bench and were inconsistent, which is to be expected, but they also had plenty of bright moments as well. Lamela is probably gone this summer which will bring an end to a very strange career for him. Lucas showed a lot of versatility this season stepping up as a CAM. He probably remains best as a substitute where his pace and dribbling can help spark the offense. Bergwijn has talent but fell out of favor under Jose and was clearly rusty when Mason worked him back into the lineup. Bale's future seems really uncertain, he seemed to enjoy his time here once Jose went bye-bye and while he isn't the world class player he was at his peak, he still has the kind of skill and class that you just can't find, and had plenty of moments despite inconsistent playing time. Spurs would surely welcome him back, and with Madrid still a tire fire, he doesn't appear to have a lot of options.
9) Kane's future is the huge question mark that will hang over this team until something is decided. Kane was the best player in the EPL this season and kept the team afloat during dark times. He reinvented himself for portions of this season as a deep-lying forward who made plays for others while also somehow managing to win the golden boot. On a team that oftentimes did not want to play attacking football. The team will not be able to find a player to replace his production if he is sold. If he is sold, will that money be immediately be reinvested in the squad during what is expected to be a difficult transfer window? If they sell Kane, they will need to spend a lot of money to shore up multiple spots on the roster to make up for the loss of his impact, and we have seen plenty of teams spend big money unwisely, such as our North London rivals. Navigating the Kane sale will be one thing, but being able to use that money to find the right players is another huge task.
Ultimately, I think the most likely path for Tottenham will be that Kane is sold, a new manager is brought in with a long-term vision, and Spurs sell off some other key players and build a younger squad that might be more of a mid-table side than a UCL side for a few years while they restock. I think Spurs have been in an interesting position since Poch first took over where the team has competed as a legitimate Top 4 side, but lack the spending power to sustain that level for forever and will need to hit the reset button. The Jose move totally backfired and the result was the club forfeited any further chances to win a major trophy with this current incarnation of the team.